Please provide a detailed explanation. Prove by induction on n that .pdf
•
0 likes•2 views
Please provide a detailed explanation. Prove by induction on n that nk =2K/(2K - 1)2(2K + 1)2 = n2 + n/(2n + 1)2.| Solution Consider for n=1 LHS=2*1/[(2-1)^2*(2+1)^2] =2/9 RHS=(1^2+1)/(2+1)^2 =2/9 Hence the given expression is true for n=1 consider the given expression is true for n hence k=1n (2k/(2k-1)^2(2k+1)^2)=n^2+n/(2n+1)^2---------->A consider for n+1 LHS= k=1n+1 (2k/(2k-1)^2(2k+1)^2) =[k=1n (2k/(2k-1)^2(2k+1)^2) ]+(2*(n+1)/(2(n+1)-1)^2*(2(n+1)+1)^2) =(n^2+n)/(2n+1)^2 +(2(n+1)/(2n+1)^2*(2n+3)^2) =[(n+1)/(2n+1)^2]*(n+(2/(2n+3)^2)) =[(n+1)/(2n+1)^2]*(n*((2n+1)+2)^2)+2)/(2n+3)^2 =[(n+1)/(2n+1)^2]*(n*(2n+1)^2+n*2*(2n+1)*2+n*4+2)/(2n+3)^2 =[(n+1)/(2n+1)^2]*(n*(2n+1)^2+n*2*(2n+1)*2+2*(2n+1))/(2n+3)^2 =[(n+1)/(2n+1)]*(n*(2n+1)+4n+2)/(2n+3)^2 =[(n+1)/(2n+1)]*(n*(2n+1)+2*(2n+1))/(2n+3)^2 =(n+1)*(n+2)/(2n+3)^2 =(n+1)(n+2)/(2n+3)^2 RHS=((n+1)^2+(n+1))/(2(n+1)+1)^2 =(n+1)(n+1+1)/(2n+3)^2 =(n+1)(n+2)/(2n+3)^2 since LHS=RHS The given expression is true for n+1 Hence by principle of mathematical induction,the given expression is true for all values of n..
Report
Share
Report
Share
Download to read offline
![Please provide a detailed explanation. Prove by induction on n that nk =2K/(2K - 1)2(2K + 1)2 =
n2 + n/(2n + 1)2.|
Solution
Consider for n=1
LHS=2*1/[(2-1)^2*(2+1)^2]
=2/9
RHS=(1^2+1)/(2+1)^2
=2/9
Hence the given expression is true for n=1
consider the given expression is true for n
hence
k=1n (2k/(2k-1)^2(2k+1)^2)=n^2+n/(2n+1)^2---------->A
consider for n+1
LHS= k=1n+1 (2k/(2k-1)^2(2k+1)^2)
=[k=1n (2k/(2k-1)^2(2k+1)^2) ]+(2*(n+1)/(2(n+1)-1)^2*(2(n+1)+1)^2)
=(n^2+n)/(2n+1)^2 +(2(n+1)/(2n+1)^2*(2n+3)^2)
=[(n+1)/(2n+1)^2]*(n+(2/(2n+3)^2))
=[(n+1)/(2n+1)^2]*(n*((2n+1)+2)^2)+2)/(2n+3)^2
=[(n+1)/(2n+1)^2]*(n*(2n+1)^2+n*2*(2n+1)*2+n*4+2)/(2n+3)^2
=[(n+1)/(2n+1)^2]*(n*(2n+1)^2+n*2*(2n+1)*2+2*(2n+1))/(2n+3)^2
=[(n+1)/(2n+1)]*(n*(2n+1)+4n+2)/(2n+3)^2
=[(n+1)/(2n+1)]*(n*(2n+1)+2*(2n+1))/(2n+3)^2
=(n+1)*(n+2)/(2n+3)^2
=(n+1)(n+2)/(2n+3)^2
RHS=((n+1)^2+(n+1))/(2(n+1)+1)^2
=(n+1)(n+1+1)/(2n+3)^2
=(n+1)(n+2)/(2n+3)^2
since LHS=RHS
The given expression is true for n+1
Hence by principle of mathematical induction,the given expression is true for all values of n.](https://image.slidesharecdn.com/pleaseprovideadetailedexplanation-230325175420-48e56dd9/85/Please-provide-a-detailed-explanation-Prove-by-induction-on-n-that-pdf-1-320.jpg)
Recommended
Mathematical Induction
1) The document uses mathematical induction to prove several formulas.
2) It demonstrates proofs for formulas like 1 + 3 + 5 + ... + (2n-1) = n^2 and 2 + 4 + ... + 2n = n(n+1).
3) The proofs follow the standard structure of mathematical induction, showing the base case is true and using the induction hypothesis to show if the statement is true for n it is also true for n+1.
1.Prove 3 n^3 + 5n , n = 1For n = 1, 1^3 + 51 = 1 + 5 = 6, a.pdf
1.
Prove 3 | n^3 + 5n , n >= 1
For n = 1, 1^3 + 5*1 = 1 + 5 = 6, and 6/3 = 2, so 3 | n^3 + 5n
Assume for n = k
3 | k^3 + 5k
Then, for k+1
(k+1)^3 + 5(k+1) = k^3 + 3k^2 + 3k + 1 + 5k + 5 = k^3 + 5k + (3k^2 + 3k + 6) = k^3 + 5k +
3(k^2 + k + 2)
From the assumption, 3 | k^3 + 5k, or k^3 + 5k = 3z for some integer z.
Thus, k^3 + 5k + 3(k^2 + k + 2) = 3z + 3(k^2 + k + 2) = 3(z + k^2 + k + 2)
Thus, 3 | n^3 + 5n for n = k+1, and the proof by induction is complete.
2. Show 2 + 5 + ...(3n - 1) = n(3n+1)/2
For n = 1, the left hand side is 2 and the right hand side is n(3n+1)/2 = 1(3*1+1)/2 = 1(4)/2 = 2
2 = 2
Assume for n = k
2 + 5 + ...(3k - 1) = k(3k+1)/2
for n = k+1
2 + 5 + ...+ (3k - 1) + (3(k+1) - 1) =
(2 + 5 + ...+ (3k - 1) ) + 3k+2 = from assumption
k(3k+1)/2 + 3k + 2 =
3k^2/2 + k/2 + 3k + 2 =
3k^2/2 + 7/2 k + 2 =
(3k^2 + 7k + 4)/2 =
(k+1)(3k+4)/2 =
(k+1)(3k+3 + 1)/2 =
(k+1)(3(k+1) + 1)/2 =
n(3n+1)/2 for n = k+1
Thus, we have proven for n=k+1 and our proof is complete.
3. Show for n >= 1, 4 | 5^n - 1
For n = 1, 5^1 - 1 = 5 - 1 = 4 and 4 | 4
Assume for n = k
4 | 5^k - 1, or 5^k - 1 = 4z for some integer z.
Then, for n = k+1
5^(k+1) - 1 = 5^(k+1) - 1 - 5^k + 5^k = 5^k - 1 + 5^(k+1) - 5^k = 5^k - 1 + 5*5^k - 5^k = 5^k - 1
+ (5-1)*5^k =
5^k - 1 + 4*5^k = from assumption, 4z + 4*5^k = 4(z + 5^k)
Thus, 4 | 5^(k+1) - 1 or 4|5^n - 1 for n = k+1, and our proof by induction is complete.
4. Show for n >= 1 7|9^n - 2^n
For n = 1 9^1 - 2^1 = 9 - 2 = 7 and 7 | 7
Assume for n = k, 7|9^k - 2^k, or 7z = 9^k - 2^k
Then, for n = k+1,
9^(k+1) - 2^(k+1) = 9*9^k - 2 *2^k = (7+2)*9^k - 2 * 2^k = 7 * 9^k + 2 * 9^k - 2 * 2^k =
7 * 9 ^ k + 2 * (9^k - 2^k) = from the assumption,
7 * 9 ^ k + 2 * 7z = 7( 9 ^ k + 14z)
Thus, 7 | 9^(k+1) - 2^(k+1), or 9^n - 2^n for n = k+1 and the proof by induction is complete.
5. 1/2 + ... + n/2^n = 2 - (n+2)/2^n
For n = 1, we have 1/2 on the left
On the right, we have 2 - (1+2)/2^1 = 2 - 3/2 = 1/2
1/2 = 1/2
Assume for n = k 1/2 + ... + k/2^k = 2 - (k+2)/2^k
Then, for n = k + 1
1/2 + ... + k/2^k + (k+1)/2^(k+1) =
(1/2 + ... + k/2^k) + (k+1)/2^(k+1) = from assumption
2 - (k+2)/2^k + (k+1)/2^(k+1) =
2 - 2(k+2)/2^(k+1) + (k+1)/2^(k+1) =
2 - (2k+4)/2^(k+1) + (k+1)/2^(k+1) =
2 - (2k+4 -(k+1))/2^(k+1) =
2 - (2k+4 - k - 1)/2^(k+1) =
2 - (k+3)/2^(k+1) =
2 - ((k+1)+2)/2^(k+1) =
2 - (n+2)/2^n for n = k+1 and the proof is complete.
Show 2^n + n <= 3^n for n >= 1 (or n >= 0
Note: for n = 0, 2^0 + 0 = 1 + 0 = 1, and 3^0 = 1, and 1 <= 1
for n = 1, 2^1 + 1 = 2 + 1 = 3, and 3^1 = 3, and 3 <= 3
Then, assume for n = k
2^k + k <= 3^k
Then, for n = k+ 1
2^(k+1) + (k+1) =
2*2^k + (k+1) =
2^k + k + 2^k + 1 <= by assumption
3^k + 2^k + 1 <= by assumption, as 1 <= k
3^k + 3^k = 2 * 3^k < as 3^k > 0
2 * 3^k + 3^k = (2+1)*3^k = 3*3^k = 3^(k+1)
Thus, 2^(k+1) + (k+1) < 3^(k+1) for k >= 1 or
2^(n) + (n) < 3^(n) for n >= 2, and
2^(n) + (n) <= 3^(n) for all natural n
B. Extended induction.
1. This problem actually does not require extended induction but can use regular ind.
Induction q
The document proves mathematical statements using induction. It shows:
1) Induction can be used to prove the summation formula 1 + 2 + 3 + ... + n = n(n+1)/2. It shows the base case of n=1 is true, and assumes the formula is true for n=k to prove it is true for n=k+1.
2) Induction can prove inequalities, like n < 2n for all positive integers n. It shows the base case is true and assumes the statement is true for n=k to prove it is true for n=k+1.
3) Induction can prove divisibility properties, like n3 - n being
Induction q
The document proves mathematical statements using induction. It shows:
1) Induction can be used to prove the summation formula 1 + 2 + 3 + ... + n = n(n+1)/2. It shows the base case of n=1 is true, and assumes the formula is true for n=k to prove it is true for n=k+1.
2) Induction can prove inequalities, like n < 2n for all positive integers n. It shows the base case is true and assumes the statement is true for n=k to prove it is true for n=k+1.
3) Induction can prove divisibility properties, like n3 - n being
Text s1 21
The document provides solutions to exercises on mathematical induction. It presents proofs for 7 formulas involving sums and sequences. Each proof follows the same structure: show the base case is true, assume the statement is true for some integer k, and prove it is true for k+1 using algebraic manipulations. This establishes the formula is true for all positive integers n by the principle of mathematical induction.
Activ.aprendiz. 3a y 3b
The document provides solutions to exercises on mathematical induction. It presents proofs for 7 formulas involving sums and sequences. Each proof follows the same structure: show the base case is true, assume the statement is true for some integer k, and prove it is true for k+1 using algebraic manipulations. This establishes the formula is true for all positive integers n by the principle of mathematical induction.
The Fundamental Solution of an Extension to a Generalized Laplace Equation
This thesis examines the fundamental solution to a generalized Laplace equation in a three-dimensional sub-Riemannian space called Grushin space. The author presents the main result as a theorem stating that the generalized Laplace operator applied to a particular function f equals zero, with f containing vector fields and functions that define the Grushin space. The proof of the theorem involves directly computing the generalized Laplace operator by taking derivatives of f with respect to the vector fields. This generalizes previous work that solved the two-dimensional case.
Further mathematics notes zimsec cambridge zimbabwe
Further mathematics notes zimsec cambridge zimbabwe advanced level A level
Zimbabwe
Zimsec
cambridge Alpro
Recommended
Mathematical Induction
1) The document uses mathematical induction to prove several formulas.
2) It demonstrates proofs for formulas like 1 + 3 + 5 + ... + (2n-1) = n^2 and 2 + 4 + ... + 2n = n(n+1).
3) The proofs follow the standard structure of mathematical induction, showing the base case is true and using the induction hypothesis to show if the statement is true for n it is also true for n+1.
1.Prove 3 n^3 + 5n , n = 1For n = 1, 1^3 + 51 = 1 + 5 = 6, a.pdf
1.
Prove 3 | n^3 + 5n , n >= 1
For n = 1, 1^3 + 5*1 = 1 + 5 = 6, and 6/3 = 2, so 3 | n^3 + 5n
Assume for n = k
3 | k^3 + 5k
Then, for k+1
(k+1)^3 + 5(k+1) = k^3 + 3k^2 + 3k + 1 + 5k + 5 = k^3 + 5k + (3k^2 + 3k + 6) = k^3 + 5k +
3(k^2 + k + 2)
From the assumption, 3 | k^3 + 5k, or k^3 + 5k = 3z for some integer z.
Thus, k^3 + 5k + 3(k^2 + k + 2) = 3z + 3(k^2 + k + 2) = 3(z + k^2 + k + 2)
Thus, 3 | n^3 + 5n for n = k+1, and the proof by induction is complete.
2. Show 2 + 5 + ...(3n - 1) = n(3n+1)/2
For n = 1, the left hand side is 2 and the right hand side is n(3n+1)/2 = 1(3*1+1)/2 = 1(4)/2 = 2
2 = 2
Assume for n = k
2 + 5 + ...(3k - 1) = k(3k+1)/2
for n = k+1
2 + 5 + ...+ (3k - 1) + (3(k+1) - 1) =
(2 + 5 + ...+ (3k - 1) ) + 3k+2 = from assumption
k(3k+1)/2 + 3k + 2 =
3k^2/2 + k/2 + 3k + 2 =
3k^2/2 + 7/2 k + 2 =
(3k^2 + 7k + 4)/2 =
(k+1)(3k+4)/2 =
(k+1)(3k+3 + 1)/2 =
(k+1)(3(k+1) + 1)/2 =
n(3n+1)/2 for n = k+1
Thus, we have proven for n=k+1 and our proof is complete.
3. Show for n >= 1, 4 | 5^n - 1
For n = 1, 5^1 - 1 = 5 - 1 = 4 and 4 | 4
Assume for n = k
4 | 5^k - 1, or 5^k - 1 = 4z for some integer z.
Then, for n = k+1
5^(k+1) - 1 = 5^(k+1) - 1 - 5^k + 5^k = 5^k - 1 + 5^(k+1) - 5^k = 5^k - 1 + 5*5^k - 5^k = 5^k - 1
+ (5-1)*5^k =
5^k - 1 + 4*5^k = from assumption, 4z + 4*5^k = 4(z + 5^k)
Thus, 4 | 5^(k+1) - 1 or 4|5^n - 1 for n = k+1, and our proof by induction is complete.
4. Show for n >= 1 7|9^n - 2^n
For n = 1 9^1 - 2^1 = 9 - 2 = 7 and 7 | 7
Assume for n = k, 7|9^k - 2^k, or 7z = 9^k - 2^k
Then, for n = k+1,
9^(k+1) - 2^(k+1) = 9*9^k - 2 *2^k = (7+2)*9^k - 2 * 2^k = 7 * 9^k + 2 * 9^k - 2 * 2^k =
7 * 9 ^ k + 2 * (9^k - 2^k) = from the assumption,
7 * 9 ^ k + 2 * 7z = 7( 9 ^ k + 14z)
Thus, 7 | 9^(k+1) - 2^(k+1), or 9^n - 2^n for n = k+1 and the proof by induction is complete.
5. 1/2 + ... + n/2^n = 2 - (n+2)/2^n
For n = 1, we have 1/2 on the left
On the right, we have 2 - (1+2)/2^1 = 2 - 3/2 = 1/2
1/2 = 1/2
Assume for n = k 1/2 + ... + k/2^k = 2 - (k+2)/2^k
Then, for n = k + 1
1/2 + ... + k/2^k + (k+1)/2^(k+1) =
(1/2 + ... + k/2^k) + (k+1)/2^(k+1) = from assumption
2 - (k+2)/2^k + (k+1)/2^(k+1) =
2 - 2(k+2)/2^(k+1) + (k+1)/2^(k+1) =
2 - (2k+4)/2^(k+1) + (k+1)/2^(k+1) =
2 - (2k+4 -(k+1))/2^(k+1) =
2 - (2k+4 - k - 1)/2^(k+1) =
2 - (k+3)/2^(k+1) =
2 - ((k+1)+2)/2^(k+1) =
2 - (n+2)/2^n for n = k+1 and the proof is complete.
Show 2^n + n <= 3^n for n >= 1 (or n >= 0
Note: for n = 0, 2^0 + 0 = 1 + 0 = 1, and 3^0 = 1, and 1 <= 1
for n = 1, 2^1 + 1 = 2 + 1 = 3, and 3^1 = 3, and 3 <= 3
Then, assume for n = k
2^k + k <= 3^k
Then, for n = k+ 1
2^(k+1) + (k+1) =
2*2^k + (k+1) =
2^k + k + 2^k + 1 <= by assumption
3^k + 2^k + 1 <= by assumption, as 1 <= k
3^k + 3^k = 2 * 3^k < as 3^k > 0
2 * 3^k + 3^k = (2+1)*3^k = 3*3^k = 3^(k+1)
Thus, 2^(k+1) + (k+1) < 3^(k+1) for k >= 1 or
2^(n) + (n) < 3^(n) for n >= 2, and
2^(n) + (n) <= 3^(n) for all natural n
B. Extended induction.
1. This problem actually does not require extended induction but can use regular ind.
Induction q
The document proves mathematical statements using induction. It shows:
1) Induction can be used to prove the summation formula 1 + 2 + 3 + ... + n = n(n+1)/2. It shows the base case of n=1 is true, and assumes the formula is true for n=k to prove it is true for n=k+1.
2) Induction can prove inequalities, like n < 2n for all positive integers n. It shows the base case is true and assumes the statement is true for n=k to prove it is true for n=k+1.
3) Induction can prove divisibility properties, like n3 - n being
Induction q
The document proves mathematical statements using induction. It shows:
1) Induction can be used to prove the summation formula 1 + 2 + 3 + ... + n = n(n+1)/2. It shows the base case of n=1 is true, and assumes the formula is true for n=k to prove it is true for n=k+1.
2) Induction can prove inequalities, like n < 2n for all positive integers n. It shows the base case is true and assumes the statement is true for n=k to prove it is true for n=k+1.
3) Induction can prove divisibility properties, like n3 - n being
Text s1 21
The document provides solutions to exercises on mathematical induction. It presents proofs for 7 formulas involving sums and sequences. Each proof follows the same structure: show the base case is true, assume the statement is true for some integer k, and prove it is true for k+1 using algebraic manipulations. This establishes the formula is true for all positive integers n by the principle of mathematical induction.
Activ.aprendiz. 3a y 3b
The document provides solutions to exercises on mathematical induction. It presents proofs for 7 formulas involving sums and sequences. Each proof follows the same structure: show the base case is true, assume the statement is true for some integer k, and prove it is true for k+1 using algebraic manipulations. This establishes the formula is true for all positive integers n by the principle of mathematical induction.
The Fundamental Solution of an Extension to a Generalized Laplace Equation
This thesis examines the fundamental solution to a generalized Laplace equation in a three-dimensional sub-Riemannian space called Grushin space. The author presents the main result as a theorem stating that the generalized Laplace operator applied to a particular function f equals zero, with f containing vector fields and functions that define the Grushin space. The proof of the theorem involves directly computing the generalized Laplace operator by taking derivatives of f with respect to the vector fields. This generalizes previous work that solved the two-dimensional case.
Further mathematics notes zimsec cambridge zimbabwe
Further mathematics notes zimsec cambridge zimbabwe advanced level A level
Zimbabwe
Zimsec
cambridge Alpro
differential-calculus-1-23.pdf
This document contains lecture notes for Engineering Mathematics - I. It covers topics in differential calculus including differential calculus I, II, and III. For each unit, it provides introductions to concepts and worked examples of finding derivatives of various functions up to the nth order. These include standard functions like exponential, trigonometric, logarithmic, and algebraic functions. Formulas for derivative rules like the product rule, quotient rule, and chain rule are also stated.
Sol86
The document discusses calculating the probability (P) that the first number in a set of numbers is the smallest. It discretizes the intervals into small units of length ε and calculates Pk, the probability the first number is smallest given it is in the kth ε-unit. Using Taylor series approximations, it derives Pk ≈ e-k2/2ε. It then approximates the sum of Pk values using an integral to find P ≈ π/2√N, where N is the number of intervals.
Sol86
The document discusses calculating the probability (P) that the first number in a set of numbers is the smallest. It discretizes the intervals into small units of length ε. Pk is defined as the probability the first number is smallest if it is in the kth ε-unit. Approximations are made to express Pk as e-k^2/2. P is then expressed as the sum of these Pk terms, which is approximated as an integral and solved to be π/2√N, where N is the total number of intervals.
Analysis Of Algorithms Ii
The document discusses recursive algorithms and recurrence relations. It provides examples of solving recurrence relations for different algorithms like Towers of Hanoi, selection sort, and merge sort. Recurrence relations define algorithms recursively in terms of smaller inputs. They are solved to find closed-form formulas for the running time of algorithms.
ゲーム理論NEXT 期待効用理論第4回 -サンクトペテルブルクのパラドックス-
The document is about infinite series and sums. It introduces common infinite series like the harmonic series and explores their properties. It shows that the sum of the harmonic series is divergent and equals positive infinity. It also examines other examples of infinite series and explores using integrals to calculate their sums.
Recurrence relation solutions
The document discusses recurrence relations and algorithms for solving recurrence relations. It begins by defining what a recurrence relation is and provides some examples of natural functions that can be expressed as recurrences. It then discusses different methods for solving recurrence relations, including iteration methods like backward substitution, substitution methods, and recursion tree methods. Specific examples are provided to demonstrate how to apply these different solving methods to common recurrence relations.
IVR - Chapter 4 - Variational methods
Heat equation. Discretization and finite difference. Explicit and implicit Euler schemes. CFL conditions. Continuous Gaussian convolution solution. Linear and non-linear scale spaces. Anisotropic diffusion. Perona-Malik and Weickert model. Variational methods. Tikhonov regularization by gradient descent. Links between variational models and diffusion models. Total-Variation regularization and ROF model. Sparsity and group sparsity. Applications to image deconvolution.
5.4 mathematical induction t
The document outlines the steps of mathematical induction:
1. Declare that an induction argument will be used
2. Verify the statement is true for the base case (usually n=1)
3. Assume the statement is true for some integer n=k
4. Use the induction assumption to prove the statement is true for n=k+1
It then provides examples demonstrating how to use induction to prove statements like sums and inequalities for all natural numbers.
1108 ch 11 day 8
Here are the key steps:
- Find the formula for the nth term (an) of an arithmetic sequence
- Plug the values given into the formula to find a and d
- Use the formula for the sum of the first n terms (Sn) of an arithmetic sequence
- Set the formula equal to the total sum given and solve for n
The goal is to set up and solve the equation systematically rather than guessing and checking numbers. Documenting the work shows the logical steps and thought process. Keep exploring new approaches to solving problems more efficiently!
International Journal of Engineering and Science Invention (IJESI)
The document introduces the concept of (k,d)-even edge-graceful labeling and investigates this labeling for odd order trees. It begins with definitions of terms and concepts related to graph labeling. The main results section defines (k,d)-even edge-graceful labeling and provides an example. Several theorems and lemmas are presented about the properties of this labeling for odd order trees, including conditions for when such a labeling does and does not exist for a given tree.
CPSC 125 Ch 2 Sec 5
This document discusses properties and solving techniques for recurrence relations. It covers linear vs nonlinear relations, homogeneous vs inhomogeneous relations, and the order of relations. It also describes the expand, guess, verify technique and using solution formulas to solve recurrence relations in closed form. Examples are provided for each concept.
Maths-MS_Term2 (1).pdf
This document provides solutions to 10 math problems from a marking scheme for Class XII. The problems cover a range of calculus and vector topics. Key steps are shown in the solutions. For example, in problem 1, integrals involving logarithmic and trigonometric functions are solved using substitution techniques. In problem 3, vectors are used to prove an identity involving the sum of two unit vectors and the angle between them. Across the solutions, various mathematical concepts are applied concisely to arrive at the answers.
Sample question paper 2 with solution
This document provides a sample question paper for Class XII Mathematics. It consists of 3 sections - Section A has 10 one-mark questions, Section B has 12 four-mark questions, and Section C has 7 six-mark questions. The paper is for 3 hours and carries a total of 100 marks. Some questions provide internal choices. Calculators are not permitted. Sample questions include finding inverse functions, evaluating integrals, solving differential equations, and probability questions.
Recurrences
The document discusses recurrences and methods for solving them. It covers:
1) Divide-and-conquer algorithms can often be modeled with recurrences. Examples include merge-sort and matrix multiplication.
2) Common methods for solving recurrences are substitution, iteration/recursion trees, and the master method. The master method provides a general solution for recurrences of the form T(n) = aT(n/b) + nc.
3) Strassen's matrix multiplication algorithm improves on the naive O(n^3) time by using a recurrence with a=7 to achieve O(n^2.81) time via the master method. Changing variables can sometimes simplify recurrences.
End sem solution
This document contains the solutions to a final exam for a communication networks course. It includes multiple problems involving calculating metrics like average transmissions, generating functions, and mean calculations for queueing models. It also contains a multicast routing problem that involves building a minimum cost tree and calculating its cost.
Quadratic Expressions
The same function can have different functional expressions in different ranges. Finding maximum/minimum values in these cases becomes very interesting
Sol58
1) There is a goal to find an approximate expression for the probability of obtaining N + x heads in 2N coin flips when N is large.
2) Using Stirling's formula and Taylor expansions, the probability is shown to approximate a Gaussian distribution: P(x) ≈ e^-x^2/N/√πN.
3) This Gaussian approximation is only valid when x is on the order of √N or less; for larger x ∼ N^3/4, higher-order terms become important.
final19
This document presents derivations of several integrals involving the error function that are contained in the table of Gradshteyn and Ryzhik. It begins by introducing the error function and some of its basic properties. It then derives recurrences and explicit formulas for integrals of the form Fn(v) = ∫v0 tn e−t2 dt. Using these results and elementary changes of variables, it evaluates several entries in the Gradshteyn and Ryzhik table. It also presents a series representation for the error function and evaluates Laplace's classical integral involving the error function.
Gr 11 equations
Quadratic equations can be solved in several ways:
1) Factorizing, by finding two numbers whose product is the constant term and sum is the coefficient of the x term.
2) Using the quadratic formula.
3) Substitution, by letting an expression like x^2 + 2x equal a variable k, and solving the simplified equation for k and back substituting.
4) Squaring both sides, but this can introduce extraneous solutions so one must check solutions.
3. Converge article.pdf
1) The document discusses how various branches of mathematics like geometry, algebra, calculus, and linear algebra converge upon a single solution related to the golden mean and complex numbers.
2) It presents equations to show how the imaginary number i, the golden mean, and the torus are mathematically linked and provide a model for the physical universe.
3) The author argues that mathematics describes the real physical world, so all branches of mathematics must converge on the same solution that matches the universe - which is modeled by a torus shape involving the golden mean.
Please list the steps of the accounting cycle.Why is this important .pdf
Please list the steps of the accounting cycle.Why is this important to the accounting profession?
Solution
Accounting cycle means the cycle by which we convert the raw data into proper financial
statements that provides information to outsiders and stakeholders and easier to understand
Steps in accounting cycle are as follows:
1) Transaction
2) Journal Entries
3) Posting in Ledger
4) Preparing trial balance
5) Worksheet
6) Adjusting journal entries
7) Preparing financial statements
8) Closing the books of accounts.
Please let me know about the material structure and properties of So.pdf
Please let me know about the material structure and properties of Solar energy. I also want
toknow the relationships of structure and properties of it.
Solution
Photovoltaic solar cells are thin silicon disks that convert sunlight into electricity. These disks act
as energy sources for a wide variety of uses, including: calculators and other small devices;
telecommunications; rooftop panels on individual houses; and for lighting, pumping, and
medical refrigeration for villages in developing countries.
The basic component of a solar cell is pure silicon, which is not pure in its natural state.To make
solar cells, the raw materials.
More Related Content
Similar to Please provide a detailed explanation. Prove by induction on n that .pdf
differential-calculus-1-23.pdf
This document contains lecture notes for Engineering Mathematics - I. It covers topics in differential calculus including differential calculus I, II, and III. For each unit, it provides introductions to concepts and worked examples of finding derivatives of various functions up to the nth order. These include standard functions like exponential, trigonometric, logarithmic, and algebraic functions. Formulas for derivative rules like the product rule, quotient rule, and chain rule are also stated.
Sol86
The document discusses calculating the probability (P) that the first number in a set of numbers is the smallest. It discretizes the intervals into small units of length ε and calculates Pk, the probability the first number is smallest given it is in the kth ε-unit. Using Taylor series approximations, it derives Pk ≈ e-k2/2ε. It then approximates the sum of Pk values using an integral to find P ≈ π/2√N, where N is the number of intervals.
Sol86
The document discusses calculating the probability (P) that the first number in a set of numbers is the smallest. It discretizes the intervals into small units of length ε. Pk is defined as the probability the first number is smallest if it is in the kth ε-unit. Approximations are made to express Pk as e-k^2/2. P is then expressed as the sum of these Pk terms, which is approximated as an integral and solved to be π/2√N, where N is the total number of intervals.
Analysis Of Algorithms Ii
The document discusses recursive algorithms and recurrence relations. It provides examples of solving recurrence relations for different algorithms like Towers of Hanoi, selection sort, and merge sort. Recurrence relations define algorithms recursively in terms of smaller inputs. They are solved to find closed-form formulas for the running time of algorithms.
ゲーム理論NEXT 期待効用理論第4回 -サンクトペテルブルクのパラドックス-
The document is about infinite series and sums. It introduces common infinite series like the harmonic series and explores their properties. It shows that the sum of the harmonic series is divergent and equals positive infinity. It also examines other examples of infinite series and explores using integrals to calculate their sums.
Recurrence relation solutions
The document discusses recurrence relations and algorithms for solving recurrence relations. It begins by defining what a recurrence relation is and provides some examples of natural functions that can be expressed as recurrences. It then discusses different methods for solving recurrence relations, including iteration methods like backward substitution, substitution methods, and recursion tree methods. Specific examples are provided to demonstrate how to apply these different solving methods to common recurrence relations.
IVR - Chapter 4 - Variational methods
Heat equation. Discretization and finite difference. Explicit and implicit Euler schemes. CFL conditions. Continuous Gaussian convolution solution. Linear and non-linear scale spaces. Anisotropic diffusion. Perona-Malik and Weickert model. Variational methods. Tikhonov regularization by gradient descent. Links between variational models and diffusion models. Total-Variation regularization and ROF model. Sparsity and group sparsity. Applications to image deconvolution.
5.4 mathematical induction t
The document outlines the steps of mathematical induction:
1. Declare that an induction argument will be used
2. Verify the statement is true for the base case (usually n=1)
3. Assume the statement is true for some integer n=k
4. Use the induction assumption to prove the statement is true for n=k+1
It then provides examples demonstrating how to use induction to prove statements like sums and inequalities for all natural numbers.
1108 ch 11 day 8
Here are the key steps:
- Find the formula for the nth term (an) of an arithmetic sequence
- Plug the values given into the formula to find a and d
- Use the formula for the sum of the first n terms (Sn) of an arithmetic sequence
- Set the formula equal to the total sum given and solve for n
The goal is to set up and solve the equation systematically rather than guessing and checking numbers. Documenting the work shows the logical steps and thought process. Keep exploring new approaches to solving problems more efficiently!
International Journal of Engineering and Science Invention (IJESI)
The document introduces the concept of (k,d)-even edge-graceful labeling and investigates this labeling for odd order trees. It begins with definitions of terms and concepts related to graph labeling. The main results section defines (k,d)-even edge-graceful labeling and provides an example. Several theorems and lemmas are presented about the properties of this labeling for odd order trees, including conditions for when such a labeling does and does not exist for a given tree.
CPSC 125 Ch 2 Sec 5
This document discusses properties and solving techniques for recurrence relations. It covers linear vs nonlinear relations, homogeneous vs inhomogeneous relations, and the order of relations. It also describes the expand, guess, verify technique and using solution formulas to solve recurrence relations in closed form. Examples are provided for each concept.
Maths-MS_Term2 (1).pdf
This document provides solutions to 10 math problems from a marking scheme for Class XII. The problems cover a range of calculus and vector topics. Key steps are shown in the solutions. For example, in problem 1, integrals involving logarithmic and trigonometric functions are solved using substitution techniques. In problem 3, vectors are used to prove an identity involving the sum of two unit vectors and the angle between them. Across the solutions, various mathematical concepts are applied concisely to arrive at the answers.
Sample question paper 2 with solution
This document provides a sample question paper for Class XII Mathematics. It consists of 3 sections - Section A has 10 one-mark questions, Section B has 12 four-mark questions, and Section C has 7 six-mark questions. The paper is for 3 hours and carries a total of 100 marks. Some questions provide internal choices. Calculators are not permitted. Sample questions include finding inverse functions, evaluating integrals, solving differential equations, and probability questions.
Recurrences
The document discusses recurrences and methods for solving them. It covers:
1) Divide-and-conquer algorithms can often be modeled with recurrences. Examples include merge-sort and matrix multiplication.
2) Common methods for solving recurrences are substitution, iteration/recursion trees, and the master method. The master method provides a general solution for recurrences of the form T(n) = aT(n/b) + nc.
3) Strassen's matrix multiplication algorithm improves on the naive O(n^3) time by using a recurrence with a=7 to achieve O(n^2.81) time via the master method. Changing variables can sometimes simplify recurrences.
End sem solution
This document contains the solutions to a final exam for a communication networks course. It includes multiple problems involving calculating metrics like average transmissions, generating functions, and mean calculations for queueing models. It also contains a multicast routing problem that involves building a minimum cost tree and calculating its cost.
Quadratic Expressions
The same function can have different functional expressions in different ranges. Finding maximum/minimum values in these cases becomes very interesting
Sol58
1) There is a goal to find an approximate expression for the probability of obtaining N + x heads in 2N coin flips when N is large.
2) Using Stirling's formula and Taylor expansions, the probability is shown to approximate a Gaussian distribution: P(x) ≈ e^-x^2/N/√πN.
3) This Gaussian approximation is only valid when x is on the order of √N or less; for larger x ∼ N^3/4, higher-order terms become important.
final19
This document presents derivations of several integrals involving the error function that are contained in the table of Gradshteyn and Ryzhik. It begins by introducing the error function and some of its basic properties. It then derives recurrences and explicit formulas for integrals of the form Fn(v) = ∫v0 tn e−t2 dt. Using these results and elementary changes of variables, it evaluates several entries in the Gradshteyn and Ryzhik table. It also presents a series representation for the error function and evaluates Laplace's classical integral involving the error function.
Gr 11 equations
Quadratic equations can be solved in several ways:
1) Factorizing, by finding two numbers whose product is the constant term and sum is the coefficient of the x term.
2) Using the quadratic formula.
3) Substitution, by letting an expression like x^2 + 2x equal a variable k, and solving the simplified equation for k and back substituting.
4) Squaring both sides, but this can introduce extraneous solutions so one must check solutions.
3. Converge article.pdf
1) The document discusses how various branches of mathematics like geometry, algebra, calculus, and linear algebra converge upon a single solution related to the golden mean and complex numbers.
2) It presents equations to show how the imaginary number i, the golden mean, and the torus are mathematically linked and provide a model for the physical universe.
3) The author argues that mathematics describes the real physical world, so all branches of mathematics must converge on the same solution that matches the universe - which is modeled by a torus shape involving the golden mean.
Similar to Please provide a detailed explanation. Prove by induction on n that .pdf (20)
International Journal of Engineering and Science Invention (IJESI)
International Journal of Engineering and Science Invention (IJESI)
More from ajayinfomatics
Please list the steps of the accounting cycle.Why is this important .pdf
Please list the steps of the accounting cycle.Why is this important to the accounting profession?
Solution
Accounting cycle means the cycle by which we convert the raw data into proper financial
statements that provides information to outsiders and stakeholders and easier to understand
Steps in accounting cycle are as follows:
1) Transaction
2) Journal Entries
3) Posting in Ledger
4) Preparing trial balance
5) Worksheet
6) Adjusting journal entries
7) Preparing financial statements
8) Closing the books of accounts.
Please let me know about the material structure and properties of So.pdf
Please let me know about the material structure and properties of Solar energy. I also want
toknow the relationships of structure and properties of it.
Solution
Photovoltaic solar cells are thin silicon disks that convert sunlight into electricity. These disks act
as energy sources for a wide variety of uses, including: calculators and other small devices;
telecommunications; rooftop panels on individual houses; and for lighting, pumping, and
medical refrigeration for villages in developing countries.
The basic component of a solar cell is pure silicon, which is not pure in its natural state.To make
solar cells, the raw materials.
Please integrate ( cos(x) sin ^2 (x) )dx a= 4pi3 b= 3pi2.pdf
Please integrate ( cos(x) / sin ^2 (x) )dx
a= 4pi/3
b= 3pi/2
NOTE: sorry my computer will not let me use the math equation program so I can\'t place in the
appropriate symbols!
Solution
put sinx =t
cosx dx =dt
dt/t^2 dt
-3/t^3
-3/(sinx)^3 (4pi/3,3pi/2)
-3/(-1)^3 +3/(-1)^3
3-3=0.
please identify true or false Solar flux or solar constant is the to.pdf
please identify true or false Solar flux or solar constant is the total amount of energy the sun
emits. Carbon dioxide is one of the six criteria pollutants regulated under Clean Air Act. Lakes
with high nutrient levels are considered oligotrophic. An ecosystem in its matured stage is more
dependent on detritus for nutrient regeneration than a developing ecosystem. Ground-level
ozone is a secondary pollutant. The impact of human activities on the environment can be
summarized as: Impact = Population x Affluence x Technology Bata (beta) radiation can travel
longer distance than alpha (alpha) particles in the air. Animal manure is an example of the
hazardous waste.
Solution
1. Solar constant is the rate at which energy reached the earth surface.
so, answer is false.
2.According to clean air act, Six pollutants are carbon dioxide, ground level ozone, particulate
matter, nitrogen dioxide, lead, sulfur dioxide. so, answer is true
3.Lake with high nutrient content is known as eutrophic lake.
So, ans is false
4.Secondary pollutants are those which are not directly emitted. so, Ground level Ozone is a
secondary pollutant
So, True
5. I= PAT is known a impact equation where I = Impact of human on the environment,
P=population, A= affluence, T= Technology
So, true (ans)
6.Beta radiation travels few times farther than alpha particles
so, ans is true.
7. Human manure is not a hazadous waste.
Answer is false.
Please help. I will rate. Find an orthonormal basis of the plane x1 .pdf
Please help. I will rate. Find an orthonormal basis of the plane x1 + 7x2-x3=0. Answer: To enter
a basis into Web Work. place the entries of each vector inside of brackets. and enter a list of
these vectors, separated by commas. For instance, if your basis is then you would enter
[1,2,3]41,1,1] into the answer blank
Solution
EQN. OF PLANE IS.
PLEASE HELP....Scenario In the five years since the midterm scenari.pdf
PLEASE HELP....Scenario: In the five years since the midterm scenario, the drainage project has
been a tremendous success and the City of Ocelot has grown rapidly as the healthier environment
has attracted new residents and businesses. The population of Ocelot is now about 40,000 with a
range of incomes from very wealthy to about 10% below the poverty level. Housing ranges from
high-end houses and condominiums to publicly subsidized apartments. The ethnic distribution of
the population is 60% Anglo-American, 20% Hispanic-American, 12% African-American, 5%
Asian-American, and 3% Native American. 1. The distinguished professors of organization
theory, Dr. Virgil Hilts and Dr. Robert Hendley, disagree over how to analyze decision-making
regarding the drainage project. Both believe the rational model is too idealistic, and Dr. Hilts
argues that the “bounded rationality” model best applies to the approach taken by the City of
Ocelot. Dr. Hendley, on the other hand, believes that Lindblom’s model of “muddling through”
offers a better explanation of how matters developed. A third opinion is offered by Dr. Roger
Bartlett, who believes that a “mixed scanning” model best explains the successful decision-
making for the young city. Compare and contrast these three approaches, and explain your view
of which model (bounded rationality, muddling through, or mixed-scanning) best applies to the
developments in Ocelot. 2. Dr. Hilts and Dr. Hendley also disagree on whether Tiebout’s model
offers much explanation for the population growth Ocelot has experienced in growing from
30,000 five years ago to the current population of 40,000. Dr. Hilts argues that residents of
neighboring communities have “voted with their feet” as Ocelot’s policies have generated a
desirable quality of life in comparison to surrounding areas and directs attention to a thriving
retirement community that includes many new arrivals. Dr. Hendley replies that the bulk of the
in-migration is younger people attracted by jobs created by new businesses that have located
offices in Ocelot. Explain which view you find more persuasive, or can the insights of these two
views be combined?
Solution
Given :
Population of Ocelot =40000, 10% below poverty line, Ethinic distribution= 60%, Anglo
ameican=20%, Hispanic american=20%,African American =5%, Asian american and 3%Native
american
Various models in consideration
Bounded rationality method can be combined with mixed scanning of the date in order for it
present a particular required output and plans for the given city.Bounded rationalityMuddling
throughMixed scanningRemarksThis is a way of rationally selectingA random way to make the
plans and execution sucessMixed scanning is a hierarchial mode of descison making that
combines higher order, fundamental decision making with lower order , incremental decision
that work out and or prepare for the higher order .Limited to information availbale and timeLess
chances of getting things accuratelyModerate c.
Please I need help with abstract algebra Will rate quicklySoluti.pdf
Please I need help with abstract algebra Will rate quickly
Solution
In algebra, which is a broad division of mathematics, abstract algebra is a common name for the
sub-area that studies algebraic structures in their own right. Such structures include groups, rings,
fields, modules, vector spaces, and algebras. The specific term abstract algebra was coined at the
beginning of the 20th century to distinguish this area from the other parts of algebra. The term
modern algebra has also been used to denote abstract algebra.
Two mathematical subject areas that study the properties of algebraic structures viewed as a
whole are universal algebra and category theory. Algebraic structures, together with the
associated homomorphisms, form categories. Category theory is a powerful formalism for
studying and comparing different algebraic structures.
History
As in other parts of mathematics, concrete problems and examples have played important roles
in the development of abstract algebra. Through the end of the nineteenth century, many --
perhaps most -- of these problems were in some way related to the theory of algebraic equations.
Major themes include:
Numerous textbooks in abstract algebra start with axiomatic definitions of various algebraic
structures and then proceed to establish their properties. This creates a false impression that in
algebra axioms had come first and then served as a motivation and as a basis of further study.
The true order of historical development was almost exactly the opposite. For example, the
hypercomplex numbers of the nineteenth century had kinematic and physical motivations but
challenged comprehension. Most theories that are now recognized as parts of algebra started as
collections of disparate facts from various branches of mathematics, acquired a common theme
that served as a core around which various results were grouped, and finally became unified on a
basis of a common set of concepts. An archetypical example of this progressive synthesis can be
seen in the history of group theory.
Early group theory
There were several threads in the early development of group theory, in modern language loosely
corresponding to number theory, theory of equations, and geometry.
Leonhard Euler considered algebraic operations on numbers modulo an integer, modular
arithmetic, in his generalization of Fermat\'s little theorem. These investigations were taken
much further by Carl Friedrich Gauss, who considered the structure of multiplicative groups of
residues mod n and established many properties of cyclic and more general abelian groups that
arise in this way. In his investigations of composition of binary quadratic forms, Gauss explicitly
stated the associative law for the composition of forms, but like Euler before him, he seems to
have been more interested in concrete results than in general theory. In 1870, Leopold Kronecker
gave a definition of an abelian group in the context of ideal class groups of a number field,
generalizing .
Please I need a help in this Q Write a sine function and a cosine fu.pdf
Please I need a help in this Q Write a sine function and a cosine function which match the
following graph. Note that the x-axis goes from 0 to 2 pi and the y-axis goes from -7 to 8.
Solution
sine function
peak to peak = 8 so one peak = 4
it\'s shifted 2 units downward , and the period is 2pi/2 = pi
f(x) = 4 sin(2x) - 2
cosine
like sine but shifted right by (pi / 4)
f(x) = 4 cos 2(x - pi/4) - 2
=> f(x) = 4 cos (2x - pi/2) - 2.
please helpA researcher is investigating the relationship betwee.pdf
please help
A researcher is investigating the relationship between
personality and birth order position. A sample of
college students is classified into four birth-order
categories (1st, 2nd, 3rd, 4th or later) and classified as
being either extroverted or introverted.
Solution
Chi-Square Test for Independence to determine whether there is a significant association
between the two variables(personality and birth order position).
please select one option Q1. HRM is associated with the manage.pdf
please select one option
Q1. HRM is associated with the management of:
A. General people
B. Financial resources
C. Organizational people
D. Community members
Q2. Which of the following management function includes settingstandards for
everyone in the organization?
A. Planning
B. Organizing
C. Leading
D. Controlling
Q3. Motivating the employees is classified as:
A. Informational role
B. Interpersonal role
C. Decisional role
D. Conceptual role
Q4. Which of the following is the main responsibility of an HRdepartment?
A. Attracting candidates for job
B. Ensure staff development
C. Keep employees motivated
D. All of the given options
Q5. HR managers are generally the _______________ managers:
A. Line
B. Middle
C. Staff
D. Top
Q6. Manufacturing was the main concern of personnel departmentduring:
A. Mechanistic period
B. Catalytic period
C. Organistic period
D. Strategic period
Q7. Running the organizational activities is called:
A. Management process
B. Executive position
C. Quality management
D. Performance measurement
Q8. Jobs are identified & grouped while:
A. Planning
B. Organizing
C. Leading
D. Controlling
Q9. Effective HRM leads to:
A. Organizational success
B. Organizational failure
C. Organizational complexity
D. Organizational inefficiency
Q10. Organizational goals should be:
A. Achievable
B. Ambiguous
C. Random
D. Vague
Q11. Customers of an organization fall under which of the followingcategory?
A. Shareholders
B. Staff
C. Partners
D. Stakeholders
Q12. What could be the best approach for an organization to sustainin a dynamic
environment?
A. Be stagnant
B. Responsive to change
C. Reluctant to change
D. Merge with others
Q13. The thorough & detailed study regarding jobs within anorganization is
represented by:
A. Job analysis
B. Job description
C. Job specification
D. Job evaluation
Q14. A practice used by companies to assign their costly activitiesto outside providers
is known as:
A. Planning
B. Decentralization
C. Restructuring
D. Outsourcing
Q15. Organizational behavior depicts the:
A. Jargons used within the organization
B. Collective behavior of employees
C. Effect of society’s behavior on an organization
D. All of the given options
Q16. Leaders perform:
A. Decisional roles
B. Informal roles
C. Informational roles
D. Interpersonal roles
Q17. Organizations take inputs from:
A. Rules & policies
B. Internal environment
C. External environment
D. Legislations & litigations
Q18. Which of the following system represents interrelatedactivities?
A. A closed system
B. An isolated system
C. An open system
D. A clogged system
Q19. Brain drain is one of the:
A. Organizational threat
B. Organizational opportunity
C. Organizational strength
D. Organizational weakness
Q20. Shifting from manual to computerized system is resulted dueto:
A. Workforce diversity
B. Technological advancement
C. Stake holders involvement
D. Globalization
Solution
1.c
2.c
3.c
4.c
5.c
6.a
7.d
8.b
9.d
10.a
11.d
12. a
13.a
14.d
15.d
16.c
17.c
18.c
19.b
20.b.
Please round 14.08 to three decimal places......SolutionTo rou.pdf
Please round 14.08 to three decimal places......
Solution
To round the decimal to three digit we need the information abount the fourth digit . If the fouth
digit is nearest to tenth digit then the third digit will rounded off to one greater and if the fouth
digit is less then 5 then the third digit will be rounded off to the digit existing.
Please reflect on the role of NPV and payback period in evaluating i.pdf
Please reflect on the role of NPV and payback period in evaluating investment alternatives. How
does this effect us as investors?
Solution
When evaluating a project or long term investment of company, it is important to reach right
decisions based on the capital budgeting methods.
Net Present Value (NPV)
For a project with one investment outlay, made initially, the net present value (NPV) is the
present value of the future after-tax cash flows minus the investment outlay.
Because the NPV is the amount by which the investor\'s wealth increases as a result of the
investment, the decision rule for the NPV is as follows:
Payback Period
The payback period is the number of years required to recover the original investment in a
project. The payback is based on cash flows. For example, if you invest $10 million in a project,
how long will it be until you recover the full original investment?
Payback period methods is simple and easy to understand, however, it has many drawbacks.
First, it is a measure of payback and not a measure of profitability. By itself the payback period
would be a dangerous criterion for evaluating capital projects.
The payback period may also be used as an indicator of project liquidity. A project with a two-
year payback may be more liquid than another project with a longer payback.
Because it is not economically sound, the payback period has no decision rule like that of the
NPV or IRR. If the payback period is being used (perhaps as a measure of liquidity), analysts
should also use an NPV or IRR to ensure that their decisions also reflect the profitability of the
projects being considered..
Please quickly Im taking the exam now.- What are the four reason.pdf
There are four main reasons for government intervention in the economy: 1) establishing equity, 2) addressing externalities, 3) managing macroeconomic instability, and 4) providing public goods. Two of these in more detail are macroeconomic stabilization, like stimulus packages during recessions, and public goods, which private sectors are unwilling to invest in due to high positive externalities.
PLease help!! 5. In a college class of 30 people, sixteen are women,.pdf
There are 30 people in a college class. 16 are women and 20 are from the East. 11 men are from the East. So there must be 4 women who are not Easterners (16 women total - 11 Eastern men = 5 non-Eastern women, and 1 of those is a man, so 4 non-Eastern women). And there must be 14 men in the class if there are 30 people total, 16 are women, and 11 Eastern men.
Please Provide details and I will raterigh.SolutionLet S = m.pdf
Please Provide details and I will raterigh.
Solution
Let |S| = m , an integer, and let |T| = n . We aregiven that m > n.
Now function f can map to one or more elements of T. Thebest case for the most number of
distinct mappings is when f coversor reaches all elements of T. In this situation, thereare m-n
overflow elements in S that map to a previouslymapped element in T (pigeon hole principle). Or
restated,there are elements s1 ands2 in S such thatf(s1) = f(s2).
If f does not cover T, then there exists some element t1 in Tthat is not reached. So, some other
element of T is reachedby one or more elements in S by f. Or, as before, thereexist one or more
elements s1, s2 of S where f(s1)=f(s2).
Please provide detailed solutions To test a proportion of defec.pdf
Please provide detailed solutions:
To test a proportion of defective items with no prior data we draw a sample Xn = (X1,..., Xn)
from a Bernoulli population with an unknown Find the likelihood function f1(x1,,xn| theta ) of
sample Find the global maximum v of fn [Find a critical point and show that it is a local
maximum point and the global maximum point of fn.] Argue that v ia an m.l.e of theta Find the
m.l.e theta of theta. Show that theta is unbiased. The number of connections to a wrong phone
number is often modeled by a Poisson distribution. Suppose we need to estimate the parameter
lambda of that distribution (the mean number of wrong connections) by observing a sample
x1,...,xn of wrong connections on n different days. Assuming that sigma = x1 + ... + xn > 0. find
the m.l.e. and MLE of lambda . Prove that the MLE of lambda is unbiased
Solution
1)fn=nCr*(p)^(r)*(1-p)^(n-r)
2)at n=r+1
i can give detailed explanatary solution of all but rate me first.
Please provide answer to question 9.2 page 231 of Manufacturing Engi.pdf
Please provide answer to question 9.2 page 231 of Manufacturing Engineering and Technologhy,
Seventh Edition.
Solution
In a composite there are two constituent materials. They are matrix and the reinforcement fibres.
Reinforcements fibres are embedded in the matrix. Fibres are generally of very high strength and
stiffness. Matirx transfer the load between the fibres. During tension fibres handle most of the
load but during compression matrix plays a major role in supporting the fibre..
Please provide a step by step and explanation. Turners syndrome is.pdf
Please provide a step by step and explanation. Turner\'s syndrome is a rare chromosomal
disorder in which girls have only one X chromosome. The condition affects about 1 in 2000 girls
in the United States. About 1 in 10 girls with Turner\'s syndrome suffers from an abnormal
narrowing of the aorta. In a group of 4000 girls, what is the probability that no girls are affected
with Turner\'s syndrome? That one girl is affected? Two? At least three? In a group of 170 girls
affected with Turner\'s syndrome, what is the probability that at least 20 of them suffer from an
abnormal narrowing of the aorta?...
Solution
We want to use Poisson\'s distribution equation for this probability = expected
number^actual number* e ^(-expected number)/actual number factorial (a) we would expect 2
out of 4000 so chance that none are affected is 2^0 (e^-2)/0! 0! = 1 2^0 = 1 e^-2 = 0.135335283
So, probability that none of the girls is affected is 0.135 or 13.5 percent One girl 2^1(e^-2)/1!
this is twice as much as we had before Probability of one girl being affected = 0.27 or 27%
Chance that two girls are affected, this is the expected number, so we are expeting a high
probability here chance that two are affected = 2^2(e^-2)/2! = 0.27 or 27% The chance that at
least three are affected is the total probability minus the chances that 0, 1, or 2 are affected
Chance of three or more = 1 - 0.135 - 0.27 - 0.27 = 0.323323584 So there is a 32% chance that
three or more will be affected. (b) now we expect 17 girls would have the narrowing of the aorta
(10% of 170 is 17). The question has a typo, but we could use the same equation. I assume you
are asking for one girl, two, and at least three. The odds for only one or two girls to have
narrowing is very low. 17^1*(e^-17)/1! = 7.03789E-07 are the odds that only one of the 170 girls
has the narrowing 17^2*(e^-17)/2! = 5.98221E-06 are the odds that only two of the 170 girls
have the narrowing The odds that at least three have the narrowing is one minus the odds that
none, one, or two have the narrowing. Odds that zero have it = 17^0*(e^-17)/0! = 4.13994E-08
So the odds that at least three have the narrowing when we expect seventeen to have the
narrowing is 1 - 4.13994E-08 - 7.03789E-07 - 5.98221E-06 = 0.999993273 In other words, there
is just about 100% chance that at least three will have the narrowing. Again, you have a typo in
(b) so you should make sure that you were asked to find this out instead of something else..
Please post your response to the following focus questionsWhat is.pdf
Please post your response to the following focus questions:
What is the difference between a felony and a misdemeanor?
What is the State
Solution
Misdemeanors are usually crimes that are considere to be not so serious, examples included but
not limited: minor drug offense, DUI, minor thefts.
Felonies in the other hand are the most serious type of crime and are classified by degrees, the
first degree being the most serious one, examples are: murder, rape. grand theft, etc.
Reference: http://www.hg.org/
The burden of proof is placed on the State, who must demonstrate that the defendant is guilty
before a jury may convict him or her. But in some jurisdiction, the defendant has the burden of
establishing the existence of certain facts that give rise to a defense, such as the insanity plea.
Reference: http://legal-dictionary.thefreedictionary.com/burden+of+proof.
More from ajayinfomatics (19)
Please list the steps of the accounting cycle.Why is this important .pdf
Please list the steps of the accounting cycle.Why is this important .pdf
Please let me know about the material structure and properties of So.pdf
Please let me know about the material structure and properties of So.pdf
Please integrate ( cos(x) sin ^2 (x) )dx a= 4pi3 b= 3pi2.pdf
Please integrate ( cos(x) sin ^2 (x) )dx a= 4pi3 b= 3pi2.pdf
please identify true or false Solar flux or solar constant is the to.pdf
please identify true or false Solar flux or solar constant is the to.pdf
Please help. I will rate. Find an orthonormal basis of the plane x1 .pdf
Please help. I will rate. Find an orthonormal basis of the plane x1 .pdf
PLEASE HELP....Scenario In the five years since the midterm scenari.pdf
PLEASE HELP....Scenario In the five years since the midterm scenari.pdf
Please I need help with abstract algebra Will rate quicklySoluti.pdf
Please I need help with abstract algebra Will rate quicklySoluti.pdf
Please I need a help in this Q Write a sine function and a cosine fu.pdf
Please I need a help in this Q Write a sine function and a cosine fu.pdf
please helpA researcher is investigating the relationship betwee.pdf
please helpA researcher is investigating the relationship betwee.pdf
please select one option Q1. HRM is associated with the manage.pdf
please select one option Q1. HRM is associated with the manage.pdf
Please round 14.08 to three decimal places......SolutionTo rou.pdf
Please round 14.08 to three decimal places......SolutionTo rou.pdf
Please reflect on the role of NPV and payback period in evaluating i.pdf
Please reflect on the role of NPV and payback period in evaluating i.pdf
Please quickly Im taking the exam now.- What are the four reason.pdf
Please quickly Im taking the exam now.- What are the four reason.pdf
PLease help!! 5. In a college class of 30 people, sixteen are women,.pdf
PLease help!! 5. In a college class of 30 people, sixteen are women,.pdf
Please Provide details and I will raterigh.SolutionLet S = m.pdf
Please Provide details and I will raterigh.SolutionLet S = m.pdf
Please provide detailed solutions To test a proportion of defec.pdf
Please provide detailed solutions To test a proportion of defec.pdf
Please provide answer to question 9.2 page 231 of Manufacturing Engi.pdf
Please provide answer to question 9.2 page 231 of Manufacturing Engi.pdf
Please provide a step by step and explanation. Turners syndrome is.pdf
Please provide a step by step and explanation. Turners syndrome is.pdf
Please post your response to the following focus questionsWhat is.pdf
Please post your response to the following focus questionsWhat is.pdf
Recently uploaded
Natural birth techniques - Mrs.Akanksha Trivedi Rama University
Natural birth techniques - Mrs.Akanksha Trivedi Rama UniversityAkanksha trivedi rama nursing college kanpur.
Natural birth techniques are various type such as/ water birth , alexender method, hypnosis, bradley method, lamaze method etcDigital Artifact 1 - 10VCD Environments Unit
Digital Artifact 1 - 10VCD Environments Unit - NGV Pavilion Concept Design
Exploiting Artificial Intelligence for Empowering Researchers and Faculty, In...
Exploiting Artificial Intelligence for Empowering Researchers and Faculty, In...Dr. Vinod Kumar Kanvaria
Exploiting Artificial Intelligence for Empowering Researchers and Faculty,
International FDP on Fundamentals of Research in Social Sciences
at Integral University, Lucknow, 06.06.2024
By Dr. Vinod Kumar KanvariaPengantar Penggunaan Flutter - Dart programming language1.pptx
Pengantar Penggunaan Flutter - Dart programming language1.pptx
Executive Directors Chat Leveraging AI for Diversity, Equity, and Inclusion
Let’s explore the intersection of technology and equity in the final session of our DEI series. Discover how AI tools, like ChatGPT, can be used to support and enhance your nonprofit's DEI initiatives. Participants will gain insights into practical AI applications and get tips for leveraging technology to advance their DEI goals.
PCOS corelations and management through Ayurveda.
This presentation includes basic of PCOS their pathology and treatment and also Ayurveda correlation of PCOS and Ayurvedic line of treatment mentioned in classics.
The Diamonds of 2023-2024 in the IGRA collection
A review of the growth of the Israel Genealogy Research Association Database Collection for the last 12 months. Our collection is now passed the 3 million mark and still growing. See which archives have contributed the most. See the different types of records we have, and which years have had records added. You can also see what we have for the future.
What is Digital Literacy? A guest blog from Andy McLaughlin, University of Ab...
What is Digital Literacy? A guest blog from Andy McLaughlin, University of Aberdeen
clinical examination of hip joint (1).pdf
described clinical examination all orthopeadic conditions .
Walmart Business+ and Spark Good for Nonprofits.pdf
"Learn about all the ways Walmart supports nonprofit organizations.
You will hear from Liz Willett, the Head of Nonprofits, and hear about what Walmart is doing to help nonprofits, including Walmart Business and Spark Good. Walmart Business+ is a new offer for nonprofits that offers discounts and also streamlines nonprofits order and expense tracking, saving time and money.
The webinar may also give some examples on how nonprofits can best leverage Walmart Business+.
The event will cover the following::
Walmart Business + (https://business.walmart.com/plus) is a new shopping experience for nonprofits, schools, and local business customers that connects an exclusive online shopping experience to stores. Benefits include free delivery and shipping, a 'Spend Analytics” feature, special discounts, deals and tax-exempt shopping.
Special TechSoup offer for a free 180 days membership, and up to $150 in discounts on eligible orders.
Spark Good (walmart.com/sparkgood) is a charitable platform that enables nonprofits to receive donations directly from customers and associates.
Answers about how you can do more with Walmart!"
Main Java[All of the Base Concepts}.docx
This is part 1 of my Java Learning Journey. This Contains Custom methods, classes, constructors, packages, multithreading , try- catch block, finally block and more.
Chapter 4 - Islamic Financial Institutions in Malaysia.pptx
Chapter 4 - Islamic Financial Institutions in Malaysia.pptxMohd Adib Abd Muin, Senior Lecturer at Universiti Utara Malaysia
This slide is special for master students (MIBS & MIFB) in UUM. Also useful for readers who are interested in the topic of contemporary Islamic banking.
The simplified electron and muon model, Oscillating Spacetime: The Foundation...
Discover the Simplified Electron and Muon Model: A New Wave-Based Approach to Understanding Particles delves into a groundbreaking theory that presents electrons and muons as rotating soliton waves within oscillating spacetime. Geared towards students, researchers, and science buffs, this book breaks down complex ideas into simple explanations. It covers topics such as electron waves, temporal dynamics, and the implications of this model on particle physics. With clear illustrations and easy-to-follow explanations, readers will gain a new outlook on the universe's fundamental nature.
RPMS TEMPLATE FOR SCHOOL YEAR 2023-2024 FOR TEACHER 1 TO TEACHER 3
RPMS Template 2023-2024 by: Irene S. Rueco
BÀI TẬP BỔ TRỢ TIẾNG ANH 8 CẢ NĂM - GLOBAL SUCCESS - NĂM HỌC 2023-2024 (CÓ FI...
BÀI TẬP BỔ TRỢ TIẾNG ANH 8 CẢ NĂM - GLOBAL SUCCESS - NĂM HỌC 2023-2024 (CÓ FI...Nguyen Thanh Tu Collection
https://app.box.com/s/y977uz6bpd3af4qsebv7r9b7s21935vdRecently uploaded (20)
Pride Month Slides 2024 David Douglas School District
Pride Month Slides 2024 David Douglas School District
Natural birth techniques - Mrs.Akanksha Trivedi Rama University
Natural birth techniques - Mrs.Akanksha Trivedi Rama University
Exploiting Artificial Intelligence for Empowering Researchers and Faculty, In...
Exploiting Artificial Intelligence for Empowering Researchers and Faculty, In...
Pengantar Penggunaan Flutter - Dart programming language1.pptx
Pengantar Penggunaan Flutter - Dart programming language1.pptx
Executive Directors Chat Leveraging AI for Diversity, Equity, and Inclusion
Executive Directors Chat Leveraging AI for Diversity, Equity, and Inclusion
What is Digital Literacy? A guest blog from Andy McLaughlin, University of Ab...
What is Digital Literacy? A guest blog from Andy McLaughlin, University of Ab...
Walmart Business+ and Spark Good for Nonprofits.pdf
Walmart Business+ and Spark Good for Nonprofits.pdf
Digital Artefact 1 - Tiny Home Environmental Design
Digital Artefact 1 - Tiny Home Environmental Design
Chapter 4 - Islamic Financial Institutions in Malaysia.pptx
Chapter 4 - Islamic Financial Institutions in Malaysia.pptx
The simplified electron and muon model, Oscillating Spacetime: The Foundation...
The simplified electron and muon model, Oscillating Spacetime: The Foundation...
RPMS TEMPLATE FOR SCHOOL YEAR 2023-2024 FOR TEACHER 1 TO TEACHER 3
RPMS TEMPLATE FOR SCHOOL YEAR 2023-2024 FOR TEACHER 1 TO TEACHER 3
BÀI TẬP BỔ TRỢ TIẾNG ANH 8 CẢ NĂM - GLOBAL SUCCESS - NĂM HỌC 2023-2024 (CÓ FI...
BÀI TẬP BỔ TRỢ TIẾNG ANH 8 CẢ NĂM - GLOBAL SUCCESS - NĂM HỌC 2023-2024 (CÓ FI...
Please provide a detailed explanation. Prove by induction on n that .pdf
- 1. Please provide a detailed explanation. Prove by induction on n that nk =2K/(2K - 1)2(2K + 1)2 = n2 + n/(2n + 1)2.| Solution Consider for n=1 LHS=2*1/[(2-1)^2*(2+1)^2] =2/9 RHS=(1^2+1)/(2+1)^2 =2/9 Hence the given expression is true for n=1 consider the given expression is true for n hence k=1n (2k/(2k-1)^2(2k+1)^2)=n^2+n/(2n+1)^2---------->A consider for n+1 LHS= k=1n+1 (2k/(2k-1)^2(2k+1)^2) =[k=1n (2k/(2k-1)^2(2k+1)^2) ]+(2*(n+1)/(2(n+1)-1)^2*(2(n+1)+1)^2) =(n^2+n)/(2n+1)^2 +(2(n+1)/(2n+1)^2*(2n+3)^2) =[(n+1)/(2n+1)^2]*(n+(2/(2n+3)^2)) =[(n+1)/(2n+1)^2]*(n*((2n+1)+2)^2)+2)/(2n+3)^2 =[(n+1)/(2n+1)^2]*(n*(2n+1)^2+n*2*(2n+1)*2+n*4+2)/(2n+3)^2 =[(n+1)/(2n+1)^2]*(n*(2n+1)^2+n*2*(2n+1)*2+2*(2n+1))/(2n+3)^2 =[(n+1)/(2n+1)]*(n*(2n+1)+4n+2)/(2n+3)^2 =[(n+1)/(2n+1)]*(n*(2n+1)+2*(2n+1))/(2n+3)^2 =(n+1)*(n+2)/(2n+3)^2 =(n+1)(n+2)/(2n+3)^2 RHS=((n+1)^2+(n+1))/(2(n+1)+1)^2 =(n+1)(n+1+1)/(2n+3)^2 =(n+1)(n+2)/(2n+3)^2 since LHS=RHS The given expression is true for n+1 Hence by principle of mathematical induction,the given expression is true for all values of n.