13 2 recursive definitions
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This document discusses recursive sequence definitions, which define each term in a sequence based on the previous term using a recursion formula. It provides examples of recursive definitions and how to write explicit definitions that directly define each term without recursion. It also gives practice problems for writing recursive and explicit definitions and applying them to population growth scenarios.
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The given document discusses arithmetic sequences and their properties. It defines an arithmetic sequence as a sequence where each term is obtained by adding a fixed number (called the common difference) to the preceding term. It provides the formula for calculating the nth term of an arithmetic sequence as an = a1 + d(n - 1), where a1 is the first term and d is the common difference. Examples are provided to determine if a sequence is arithmetic or not based on this definition and formula. The document also contains practice problems asking users to find missing terms, identify patterns, and calculate specific terms of arithmetic sequences.
A Positive Integer 𝑵 Such That 𝒑𝒏 + 𝒑𝒏+𝟑 ~ 𝒑𝒏+𝟏 + 𝒑𝒏+𝟐 For All 𝒏 ≥ 𝑵
According to Bertrand's postulate, we have 𝑝𝑛 + 𝑝𝑛 ≥ 𝑝𝑛+1. Is it true that for all 𝑛 > 1 then 𝑝𝑛−1 + 𝑝𝑛 ≥𝑝𝑛+1? Then 𝑝𝑛 + 𝑝𝑛+3 > 𝑝𝑛+1 + 𝑝𝑛+2where 𝑛 ≥ 𝑁, 𝑁 is a large enough value?
patterns_and_sequences_1.ppt
The document discusses patterns and sequences. It explains that a sequence is a pattern of numbers and defines key terms like the first term, second term, and nth term. It provides examples of arithmetic sequences where the difference between consecutive terms is constant and shows how to find the nth term of an arithmetic sequence using a formula. The document also introduces quadratic sequences where the difference changes and explains how to find the formula for the nth term of a quadratic sequence.
11.3 Geometric Sequences and Series
Find the nth term of a sequence
Find the index of a given term of a sequence
Given a geometric series, be able to calculate the nth partial sum
Identify a geometric series as convergent or divergent.
Lecture 5 6_7 - divide and conquer and method of solving recurrences
The document discusses divide and conquer algorithms and solving recurrences. It covers asymptotic notations, examples of divide and conquer including finding the largest number in a list, recurrence relations, and methods for solving recurrences including iteration, substitution, and recursion trees. The iteration method involves unfolding the recurrence into a summation. The recursion tree method visually depicts recursive calls in a tree to help solve the recurrence. Divide and conquer algorithms break problems into smaller subproblems, solve the subproblems recursively, and combine the solutions.
Grade 10 Math Module 1 searching for patterns, sequence and series
This module introduces sequences and their different types. It discusses finding patterns in sequences to determine the next term. Specific examples are provided to demonstrate writing the first few terms of a sequence given its general formula. The key concepts covered are defining sequences, finite vs infinite sequences, terms of a sequence, increasing vs decreasing sequences, and using the general formula to find specific terms. Students are expected to be able to list terms of sequences, derive the formula, generate terms recursively, and describe arithmetic sequences in different ways.
Recurrence relation
A recurrence relation defines a sequence based on a rule that gives the next term as a function of previous terms. There are three main methods to solve recurrence relations: 1) repeated substitution, 2) recursion trees, and 3) the master method. Repeated substitution repeatedly substitutes the recursive function into itself until it is reduced to a non-recursive form. Recursion trees show the successive expansions of a recurrence using a tree structure. The master method provides rules to determine the time complexity of divide and conquer recurrences.
Arithmetic Sequence and Arithmetic Series
The document provides information about arithmetic sequences and arithmetic series. It defines an arithmetic sequence as a sequence of numbers where each term after the first is obtained by adding the same constant to the previous term. It gives examples of arithmetic sequences and explains how to find the common difference, the nth term of a sequence using the general formula, and how to solve problems involving arithmetic sequences and series. The last paragraph tells a story about how Carl Friedrich Gauss was able to quickly calculate the sum of all numbers from 1 to 100 by recognizing it as an arithmetic series.
recurence solutions
The document discusses recurrence relations and methods for solving them. It defines a recurrence relation as an equation that expresses the terms of a sequence in terms of previous terms. It provides examples of homogeneous and non-homogeneous recurrence relations. For homogeneous relations, it describes guessing a solution of the form T(n)=x^n and finding the characteristic equation. For non-homogeneous relations, it explains adding the non-recursive term to the characteristic equation. It then works through examples of solving both homogeneous and non-homogeneous recurrence relations.
Similar to 13 2 recursive definitions (20)
A Positive Integer 𝑵 Such That 𝒑𝒏 + 𝒑𝒏+𝟑 ~ 𝒑𝒏+𝟏 + 𝒑𝒏+𝟐 For All 𝒏 ≥ 𝑵
A Positive Integer 𝑵 Such That 𝒑𝒏 + 𝒑𝒏+𝟑 ~ 𝒑𝒏+𝟏 + 𝒑𝒏+𝟐 For All 𝒏 ≥ 𝑵
Lecture 5 6_7 - divide and conquer and method of solving recurrences
Lecture 5 6_7 - divide and conquer and method of solving recurrences
Grade 10 Math Module 1 searching for patterns, sequence and series
Grade 10 Math Module 1 searching for patterns, sequence and series
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7.3 rational exponents
Nth roots can be expressed using rational exponents, where the denominator of the exponent is the index of the radical. To evaluate expressions with rational exponents, one should rewrite them in radical form, evaluate the radical, and then raise the expression to the power indicated by the original exponent. Examples are provided of evaluating various expressions with rational exponents.
6.6 quadratic formula
The document introduces the quadratic formula as a method for solving any quadratic equation in the standard form of ax^2 + bx + c = 0. It provides the quadratic formula, and shows an example of using it to solve for x in a quadratic equation. Users are then prompted to try solving for x in a quadratic equation using the quadratic formula.
6 7 new look at conics
This document discusses classifying second-degree equations as different types of conic sections. It provides the general form of a second-degree equation, explains how to use the discriminant to determine if the graph is a circle, ellipse, parabola, or hyperbola, and gives examples of identifying the conic section from different equations. Key aspects covered include degenerate conics, the general form containing terms for x2, xy, y2, x, y, and the constant, and using the discriminant calculated from the equation coefficients to classify the conic section.
6 6 systems of second degree equations
This document discusses methods for solving systems of second-degree equations, including algebraic substitution or elimination methods and graphical methods. It provides 3 examples of solving systems of equations algebraically and graphically, including solving systems of quadratic equations and systems involving circles.
4.4 multi step trig problems
This document provides instructions for solving for the length of a shared side (x) of two triangles. It instructs the user to write two equations, one for each triangle, relating the lengths of the sides including x. It also describes an approach for solving when the triangles form one large non-right triangle, which is to draw the altitude splitting it into two triangles and using the fact that triangle angles sum to 180 degrees. The user is then told to solve the two equations to find the value of x.
6 5 parabolas
This document provides an overview of parabolas including their key properties and examples of how to graph and write equations of parabolas. It defines that every point on a parabola is equidistant from the focus and directrix, with the directrix perpendicular to the line of symmetry and the vertex halfway between them. Examples are given of finding the focus, directrix, and graphing parabolas given these features as well as writing the equation of a parabola given its directrix.
4.3 finding missing angles
Inverse trig ratios, including sin-1, cos-1, and tan-1, are used to solve for missing angle measures. When given a decimal value for sin, cos, or tan, the corresponding inverse trig function is used to find the angle measure to the nearest degree. When given a triangle, the appropriate trig ratio is set up and the inverse function is then used to find the measure of the indicated angle to the nearest degree.
4.2 solving for missing sides
This document provides instructions for solving for a missing side of a triangle using trigonometric functions. It tells the reader to label the sides of the triangle, set up an equation using the information given, and solve for the variable to find the missing side, making sure a calculator is in degree mode.
4.1 trig ratios
This document defines and explains the three main trigonometric ratios - sine, cosine, and tangent - which are used to solve for missing angles and sides of right triangles. It identifies the hypotenuse as the longest side, and the adjacent and opposite sides in relation to a given angle. The document provides the ratio equations and examples of calculating trig ratios for various angles in a right triangle.
R.4 solving literal equations
This document defines a literal equation as an equation with two or more different variables. It provides examples of literal equations such as A = bh, where b is solved for, and P = 2L + 2W, where W is solved for. The document demonstrates how to solve various literal equations for the indicated variable, such as solving for P in I = Prt, solving for x in A = x + y^2, and solving for x and y in two additional equations.
R.3 solving 1 var inequalities
This document provides instructions for solving inequalities and graphing their solutions. It notes that when multiplying or dividing an inequality by a negative number, the inequality symbol is reversed. Open circles are used for < and >, while closed circles are used for ≤ and ≥. Examples are provided of solving various inequalities algebraically and graphing the resulting solution sets on a number line.
R.2 solving multi step equations
Multi-step equations involve more than two steps to solve. To solve them, simplify the side with variables by distributing and combining like terms, then solve as a single-step or two-step equation. The key steps are to distribute any parentheses, collect like variable terms on one side and constants on the other, and use opposite operations to solve for the variable.
R.1 simplifying expressions
Algebraic expressions contain numbers, variables, and operators and can be simplified using the distributive property and by combining like terms. The document provides examples of algebraic expressions that can be simplified by first distributing all terms inside parentheses and then combining like terms, such as 3(4n + 2) + n, which is simplified to 12n + 6 + n = 13n + 6.
10 4 solving trig equations
This document discusses two methods for solving trigonometric equations: graphing and algebraically. Graphing allows for visually seeing solutions but may result in decimal answers, while algebraic solving provides exact solutions but can become messy. It provides tips for algebraic solving based on whether the equation contains functions of 2x and x, or only 2x. Examples are given to demonstrate each method.
7 2 adding and subtracting polynomials
The document discusses several methods for adding and subtracting polynomials: using algebra tiles, the horizontal method, and the vertical method. It provides examples of adding and subtracting polynomials with one and two variables. Terms with the same variables are combined by adding the coefficients. For subtraction, the signs of the terms in the subtracted polynomial are changed before adding.
7 3 multiplying polynomials
This document discusses four methods for multiplying binomial expressions: algebra tiles, the distributive property, a table or "box" method, and FOIL. It explains that every term in the first binomial must be multiplied by every term in the second binomial. Examples are provided of multiplying binomials such as (x + 2)(x + 5), along with practice problems like (x + 3)(x - 4) and (2p - 4)(3p + 2).
7.2 simplifying radicals
The document discusses simplifying radical expressions. It defines key terms like the radical sign and radicand. It provides two methods for simplifying radicals: using the product property to rewrite with the largest perfect square factor or making a factor tree to pull out factors. Examples are provided to demonstrate simplifying radicals of various forms, including those with variables and higher root expressions. The document also contains practice problems for simplifying radical expressions.
10 3 double and half-angle formulas
1) This document discusses double-angle and half-angle formulas for trigonometric functions like sine, cosine, and tangent.
2) It derives formulas that relate trig functions of double and half angles to trig functions of the original angle, such as sin(2x) = 2sin(x)cos(x) and sin(x/2) = ±√(1-cos(x))/2.
3) Examples are provided to demonstrate applying these formulas to simplify trigonometric expressions and derive new identities.
6.7 other methods for solving
This document discusses two additional methods for solving quadratic equations: the quadratic formula and graphing. The quadratic formula can be used to solve any quadratic equation in standard form (ax^2 + bx + c = 0) by inputting the a, b, and c coefficients. Graphing involves changing the equation to y= form, plotting it on a calculator to find the x-intercepts, which are the solutions. Both methods are demonstrated with examples.
10 2 sum and diff formulas for tangent
1) Formulas are derived for tangent of the sum and difference of two angles using previous trigonometric identities: tan(α + β) = (tanα + tanβ) / (1 - tanαtanβ) and tan(α - β) = (tanα - tanβ) / (1 + tanαtanβ).
2) These formulas are valid when the tangents of the individual angles and their sum/difference are defined, which excludes cases where the cosine of any angle is 0.
3) Examples are provided to demonstrate applying these formulas to find the exact value of tangents of sums/differences of known angles.
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13 2 recursive definitions
- 1. Objectives: 1. Write and use recursive sequence definitions.
- 2. What comes next? 3, 7, 11, 15, … How do you know?
- 3. Recursive Definitions Give value of tn in terms of previous term tn – 1. Contain two parts: Initial condition tells where it starts Recursion formula tells how each term is related to previous term
- 4. Example t1 = 3 tn = 2tn – 1 + 1 t2 = t3 = t4 =
- 5. You Try! Find the first 5 terms: t1 = 10 tn = tn – 1 + n
- 6. Example Give a recursive definition for: 2, 6, 18, 54, 162, … 1, 2, 6, 24, 120, 720, …
- 7. You Try! Give a recursive definition for: 81, 27, 9, 3, …
- 8. Recursive vs. Explicit 23, 20, 17, 14, … Recursive definition: t1 = 23 tn = tn – 1 – 3 Explicit definition: tn = 26 – 3n
- 9. Finding Explicit Definitions Find the first 5 terms of t1 = 3 tn = tn – 1 + 4 Write an explicit definition.
- 10. You Try! Write an explicit definition for t1 = 8 tn = ½ tn – 1
- 11. Applications The population of some country grows two ways: 1. Annual growth of 1% 2. 20,000 people immigrate each year If the population is now 5,000,000, what will it be in 4 years?