Arithmeticprogression 130714002550-phpapp02
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1) An arithmetic progression (AP) is a sequence of numbers where the difference between consecutive terms is constant. 2) The general formula for the nth term of an AP is an = a + (n-1)d, where a is the first term and d is the common difference. 3) To find the sum of the first n terms of an AP, the formula is Sn = n/2 * [2a + (n-1)d], where a is the first term, d is the common difference, and n is the number of terms.
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Arithmetic progression
An arithmetic progression (AP) is a sequence of numbers where the difference between consecutive terms is constant. The key characteristics of an AP include:
- The common difference (d) is the constant value added between terms
- The nth term (an) can be calculated as the first term (a) plus (n-1) times the common difference (d)
- The sum of the first n terms (Sn) can be calculated using the formula n/2[2a + (n - 1)×d]
Formulas are provided to calculate the nth term, sum of terms, and sum when given the first and last terms in an AP.
Arithmetic progressions
An arithmetic progression is a sequence of numbers where each term is calculated by adding a constant value, called the common difference, to the previous term. The common difference is the fixed amount subtracted between any two consecutive terms. The general formula for an arithmetic progression is an = a + (n-1)d, where a is the first term and d is the common difference. Some key points covered are:
- Sequences have a specific relation between consecutive terms
- Examples show calculating subsequent terms by adding the common difference
- The formula is used to find specific terms like the 5th term
- The sum of n terms can be calculated using a formula of n/2 * (2a + (n-1
Arithmetic progression
This PPT will clarify your all doubts in Arithmetic Progression.
Please download this PPT and if any doubt according to this PPT, please comment , then i will try to solve your problem.
Thank you :)
Arithmetic Progression
This document defines and provides examples of arithmetic progressions. An arithmetic progression is a sequence of numbers where each term is calculated by adding a fixed number, called the common difference, to the preceding term. The document provides formulas for calculating the nth term and sum of terms in an arithmetic progression. Examples are given for finding specific terms, number of terms, common differences, and sums of arithmetic progressions.
Arithmatic progression for Class 10 by G R Ahmed
An arithmetic progression (AP) is a sequence of numbers where the difference between consecutive terms is constant. The constant difference is called the common difference. The sum of the first n terms of an AP can be calculated using the formula: Sum = n/2 * (first term + last term). Several word problems are presented involving finding sums, terms, or properties of AP sequences.
Arithmetic progression
This document defines an arithmetic progression as a sequence of numbers where each term is calculated by adding a fixed number (the common difference) to the preceding term. It provides the formulas to calculate the nth term and the sum of the first n terms of an arithmetic progression given the first term, common difference, and n. It also works through examples of finding the 12th term when the 1st term is 2 and common difference is 2, and finding the sum of the first 10 terms when the 1st term is 5 and common difference is 3.
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An arithmetic progression is a sequence of numbers where the difference between consecutive terms is constant. The document provides examples of arithmetic progressions and discusses formulas for calculating the nth term in a sequence, the sum of terms in a finite arithmetic progression, and solving problems involving finding the number of terms, first term, or common difference given values in the sequence.
CBSE Class XI Maths Arthmetic progression
An arithmetic progression is a sequence of numbers where each term after the first is calculated by adding a fixed number, called the common difference, to the previous term. The nth term can be calculated as an = a + (n - 1)d, where a is the first term and d is the common difference. An arithmetic progression can be either finite, with a fixed number of terms, or infinite, with an unlimited number of terms. The sum of the first n terms of an arithmetic progression is given by Sn = n/2(2a + (n-1)d).
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Arithmetic progression
An arithmetic progression (AP) is a sequence of numbers where the difference between consecutive terms is constant. The key characteristics of an AP include:
- The common difference (d) is the constant value added between terms
- The nth term (an) can be calculated as the first term (a) plus (n-1) times the common difference (d)
- The sum of the first n terms (Sn) can be calculated using the formula n/2[2a + (n - 1)×d]
Formulas are provided to calculate the nth term, sum of terms, and sum when given the first and last terms in an AP.
Arithmetic progressions
An arithmetic progression is a sequence of numbers where each term is calculated by adding a constant value, called the common difference, to the previous term. The common difference is the fixed amount subtracted between any two consecutive terms. The general formula for an arithmetic progression is an = a + (n-1)d, where a is the first term and d is the common difference. Some key points covered are:
- Sequences have a specific relation between consecutive terms
- Examples show calculating subsequent terms by adding the common difference
- The formula is used to find specific terms like the 5th term
- The sum of n terms can be calculated using a formula of n/2 * (2a + (n-1
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This PPT will clarify your all doubts in Arithmetic Progression.
Please download this PPT and if any doubt according to this PPT, please comment , then i will try to solve your problem.
Thank you :)
Arithmetic Progression
This document defines and provides examples of arithmetic progressions. An arithmetic progression is a sequence of numbers where each term is calculated by adding a fixed number, called the common difference, to the preceding term. The document provides formulas for calculating the nth term and sum of terms in an arithmetic progression. Examples are given for finding specific terms, number of terms, common differences, and sums of arithmetic progressions.
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An arithmetic progression (AP) is a sequence of numbers where the difference between consecutive terms is constant. The constant difference is called the common difference. The sum of the first n terms of an AP can be calculated using the formula: Sum = n/2 * (first term + last term). Several word problems are presented involving finding sums, terms, or properties of AP sequences.
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- An AP is a sequence of numbers where each subsequent term is calculated by adding a fixed number (the common difference) to the previous term.
- The general term of an AP is expressed as an + (n-1)d, where a is the first term and d is the common difference.
- The sum of the first n terms of an AP is calculated as Sn = (n/2)(a + l), where a is the first term, l is the last term, and n is the number of terms being summed.
Geometric Progressions
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This document defines arithmetic progressions and provides formulas to calculate the nth term and sum of the first n terms of an arithmetic progression. It states that an arithmetic progression is a list of numbers where each subsequent term is obtained by adding a fixed number, called the common difference, to the preceding term. Formulas are derived to calculate the nth term as an = a + (n-1)d, where a is the first term and d is the common difference. The formula to calculate the sum of the first n terms is derived as Sn = n/2(a + an), where a is the first term, an is the nth term, and n is the number of terms. Some examples are provided to demonstrate using the
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For class 10.
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1. The document contains examples of arithmetic progressions and their properties. It includes problems involving finding terms, common differences, and sums of APs.
2. One problem involves two APs where the ratio of the sum of the first n terms is 7n+1:4n+27. It is shown that the ratio of the 11th terms is 4:3.
3. Another problem proves that if an AP has (2n+1) terms, the ratio of the sum of odd terms to the sum of even terms is (n+1):n.
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Arithmetic progression ex no. 4
1. The sum of the first 20 natural numbers is 210.
2. The sum of all odd natural numbers from 1 to 150 is 5625.
3. The sum of the terms of an AP with a = 6 and d = 3 up to the 10th term is 195.
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The document discusses arithmetic progressions (AP), which are lists of numbers where each subsequent term is calculated by adding a fixed number (the common difference) to the previous term. It provides examples of APs and explains how to determine if a list is an AP based on having a common difference between terms. The key characteristics of an AP include the first term, common difference, a formula for calculating the nth term, and a formula for calculating the sum of the first n terms. Examples are provided to demonstrate how to use the formulas to find the 10th term or sum of terms in a given AP.
Geometric progressions
The document defines geometric progressions as sequences where the ratio between successive terms is constant. It provides examples of geometric progressions and expresses the general form as a, ar, ar2, ar3, etc., where a is the first term and r is the common ratio. Formulas are given for the nth term and sum of n terms of a geometric progression. Several examples are then worked through applying these formulas to find specific terms and sums of terms for given progressions.
Arithmetic progression
The document discusses arithmetic progressions (AP), which are sequences where the difference between successive terms is constant. It defines an AP using the notation an = a + (n-1)d, where a is the first term, d is the common difference, and n indexes the terms. It provides examples and explains how to find individual terms, determine if a sequence is an AP, and calculate the sum of terms in an AP using the formula Sn = 1/2n(2a + (n-1)d). Various problems are worked through to demonstrate applying the concepts and formulas for APs.
Arithmetic progression
- An arithmetic progression is a sequence of numbers where the difference between consecutive terms is constant. It can be represented by the first term (a), common difference (d), and last term (l).
- The general formula for the nth term (Tn) of an AP is: Tn = a + (n-1)d
- The sum (Sn) of the first n terms of an AP can be calculated as: Sn = (n/2)(a + l)
- Inserting arithmetic means between two numbers a and b results in an AP where the common difference (d) is (b-a)/(n+1) and the inserted terms are: a + d, a + 2
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A geometric progression is a series where each term is found by multiplying the previous term by a fixed number called the common ratio. The nth term is given by anrn-1 and the sum of the first n terms is given by (1-rn)/(1-r). Key formulas are provided to calculate individual terms and the partial sum using the first term a, common ratio r, and number of terms n. Examples demonstrate applying the formulas to find terms, partial sums, common ratios, and first terms for a variety of geometric series.
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Ppt formula for sum of series
- An arithmetic sequence has the form a, a + d, a + 2d, a + 3d, etc, where d is the common difference
- The nth term is given by tn = a + (n - 1)d
- The sum of n terms of an arithmetic series is given by either Sn = (a + l)/2 or Sn = (2a + (n - 1)d)/2, where l is the last term
- Examples are provided to demonstrate calculating the nth term, last term, and sum of an arithmetic sequence given relevant information like the first term, common difference, and number of terms
Ani agustina (a1 c011007) polynomial
The document describes aspects of polynomials including:
- Definitions of polynomials and their components
- Algebraic operations on polynomials like addition, subtraction, and multiplication
- Dividing polynomials using long division and synthetic (Horner's) division methods
- The remainder theorem stating the remainder of dividing a polynomial by (x - k) is the value of the polynomial at k
- The factorization theorem relating the factors of a polynomial to its roots
- Techniques for factorizing polynomials based on the sums of coefficients
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aapp.pdf
- An arithmetic progression (AP) is a sequence of numbers where the difference between consecutive terms is constant.
- The general form of an AP is: a, a + d, a + 2d, a + 3d, ..., a + (n - 1)d
- Where a is the first term and d is the common difference, which is the amount each term increases by.
- To find the nth term, the formula is: an = a + (n - 1)d
- The sum of the first n terms of an AP can be calculated as: Sn = n/2 * (2a + (n - 1)d)
Arithmetic-Progressions-4.pptxhejejejejejekkee
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This document defines arithmetic progressions and provides examples to illustrate key concepts such as common difference, general term, and formulas to calculate the sum of terms. It includes 10 practice problems with solutions to find missing terms, sums of terms, and numbers in arithmetic progressions given information such as terms, sums, and products.
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- The general term of an AP is expressed as an + (n-1)d, where a is the first term and d is the common difference.
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2. One problem involves two APs where the ratio of the sum of the first n terms is 7n+1:4n+27. It is shown that the ratio of the 11th terms is 4:3.
3. Another problem proves that if an AP has (2n+1) terms, the ratio of the sum of odd terms to the sum of even terms is (n+1):n.
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This document discusses arithmetic and geometric progressions. It defines arithmetic and geometric sequences as lists of numbers where each subsequent term is calculated using a common difference or ratio. It provides formulas to calculate the nth term and sum of the first n terms for both progressions. The document also discusses arithmetic and geometric means as the averages between two numbers in an arithmetic or geometric progression.
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The document discusses arithmetic progressions (AP), which are lists of numbers where each subsequent term is calculated by adding a fixed number (the common difference) to the previous term. It provides examples of APs and explains how to determine if a list is an AP based on having a common difference between terms. The key characteristics of an AP include the first term, common difference, a formula for calculating the nth term, and a formula for calculating the sum of the first n terms. Examples are provided to demonstrate how to use the formulas to find the 10th term or sum of terms in a given AP.
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The document discusses arithmetic progressions (AP), which are sequences where the difference between successive terms is constant. It defines an AP using the notation an = a + (n-1)d, where a is the first term, d is the common difference, and n indexes the terms. It provides examples and explains how to find individual terms, determine if a sequence is an AP, and calculate the sum of terms in an AP using the formula Sn = 1/2n(2a + (n-1)d). Various problems are worked through to demonstrate applying the concepts and formulas for APs.
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- The sum (Sn) of the first n terms of an AP can be calculated as: Sn = (n/2)(a + l)
- Inserting arithmetic means between two numbers a and b results in an AP where the common difference (d) is (b-a)/(n+1) and the inserted terms are: a + d, a + 2
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Ppt formula for sum of series
- An arithmetic sequence has the form a, a + d, a + 2d, a + 3d, etc, where d is the common difference
- The nth term is given by tn = a + (n - 1)d
- The sum of n terms of an arithmetic series is given by either Sn = (a + l)/2 or Sn = (2a + (n - 1)d)/2, where l is the last term
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The document describes aspects of polynomials including:
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- Algebraic operations on polynomials like addition, subtraction, and multiplication
- Dividing polynomials using long division and synthetic (Horner's) division methods
- The remainder theorem stating the remainder of dividing a polynomial by (x - k) is the value of the polynomial at k
- The factorization theorem relating the factors of a polynomial to its roots
- Techniques for factorizing polynomials based on the sums of coefficients
The document contains examples and explanations of key polynomial concepts over multiple pages.
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Similar to Arithmeticprogression 130714002550-phpapp02
aapp.pdf
- An arithmetic progression (AP) is a sequence of numbers where the difference between consecutive terms is constant.
- The general form of an AP is: a, a + d, a + 2d, a + 3d, ..., a + (n - 1)d
- Where a is the first term and d is the common difference, which is the amount each term increases by.
- To find the nth term, the formula is: an = a + (n - 1)d
- The sum of the first n terms of an AP can be calculated as: Sn = n/2 * (2a + (n - 1)d)
Arithmetic-Progressions-4.pptxhejejejejejekkee
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arithmatic progression.pptx
The document defines and provides examples of arithmetic progressions (AP). It states that an AP is a sequence where the difference between consecutive terms is constant. The key characteristics of an AP include:
- The first term is denoted as a
- The common difference is denoted as d
- The nth term is calculated as an = a + (n-1)d
Several examples are provided to illustrate how to determine if a sequence is an AP and how to calculate individual terms and the sum of terms in an AP using the above formulae.
Sequence and series
SEQUENCE AND SERIES
SEQUENCE
Is a set of numbers written in a definite order such that there is a rule by which the terms are obtained. Or
Is a set of number with a simple pattern.
Example
1. A set of even numbers
• 2, 4, 6, 8, 10 ……
2. A set of odd numbers
• 1, 3, 5, 7, 9, 11….
Knowing the pattern the next number from the previous can be obtained.
Example
1. Find the next term from the sequence
• 2, 7, 12, 17, 22, 27, 32
The next term is 37.
2. Given the sequence
• 2, 4, 6, 8, 10, 12………
Arithmetic progression
The document defines an arithmetic progression (AP) as a sequence of numbers where each term is calculated by adding a constant value, called the common difference, to the preceding term. The general formula for calculating any term in an AP is: tn = a + (n - 1)d, where a is the first term, d is the common difference, and n is the term number. Several examples of AP sequences are provided to illustrate the definition and formula. Methods for calculating the nth term, sum of terms, and finding consecutive terms in an AP are also explained.
AP&GP.pptx
The document discusses arithmetic progressions (AP) and geometric progressions (GP). It defines an AP as a sequence where the difference between consecutive terms is constant, and provides the formulas for the nth term, sum of n terms, and examples of finding terms and sums. A GP is defined as a sequence where each term is the previous term multiplied by a common ratio, and the formulas for the nth term and sum of n terms are given, along with examples. Various word problems demonstrate calculating terms and sums for APs and GPs.
Series in Discrete Structure || Computer Science
The document defines and provides examples of sequences, series, arithmetic progressions, and geometric progressions. It gives the formal definitions and notations used to represent these concepts. Several examples are worked through to demonstrate finding explicit formulas for sequences based on given terms, computing specific terms of sequences, and identifying terms of sequences based on given properties like the first term, common difference or ratio, and a target term value.
Lesson 1- Math 10 - W1Q1_ Arithmetic Sequences and Series.pptx
Here are the steps to solve these problems:
1) Given: a1 = 15, d = -6, n = 73
Use the formula: an = a1 + (n - 1)d
a73 = 15 + (73 - 1)(-6) = 15 - 432 = -417
2) Given: a1 = -1, d = 5, n = 28
Use the formula: an = a1 + (n - 1)d
a28 = -1 + (28 - 1)(5) = -1 + 135 = 134
3) Given: a1 = 5 seats, d = 3 additional seats, n = 42 rows
Use the formula: an
Arithmeticprogression
An arithmetic progression is a sequence of numbers where the difference between consecutive terms is constant. The nth term (an) of an arithmetic progression can be calculated as an = a + (n-1)d, where a is the first term and d is the common difference. The sum (Sn) of the first n terms of an arithmetic progression is calculated as Sn = n/2 * (2a + (n-1)d).
Math aruthmetic geometric series
The document discusses summation formulas for arithmetic and geometric series. It provides the formulas to calculate the sum of a finite arithmetic series as the mean of the first and last term multiplied by the number of terms. For a geometric series with a common ratio r not equal to 1 and divisor not equal to 0, the formula is the first term divided by 1 minus the common ratio. An example calculates the 30th term of an arithmetic series with initial term of 4 and difference of 3 between terms. It also finds a rule for the nth term of a geometric sequence given the first two terms.
Sequences And Series
Gauss solved his teacher's challenge of summing the integers from 1 to 100 immediately by realizing that the normal sum (1 + 2 + ... + 100) and the reverse sum (100 + 99 + ... + 1) would be equal. Adding these sums together results in 101 multiplied by the number of terms (100), giving the solution of 5050. This insight showed that mechanical solving is not as useful as understanding the underlying patterns in mathematical problems.
Presentation of Arithmatic sequence Series Created by Ambreen koondhar.pptx
The document presents on the topic of arithmetic sequences and series. It defines an arithmetic sequence as a sequence where the differences between successive terms are the same. An arithmetic series is defined as the sum of terms in an arithmetic sequence. It provides an example of an arithmetic sequence where each term is obtained by adding 3 to the previous term. The document also presents the formulas for finding the nth term and the sum of the first n terms (arithmetic series) of a sequence where the first term is a and the common difference is d. It provides an example problem demonstrating how to use the formula to calculate the sum of earnings over the first 5 years given the initial earning and yearly increase amounts.
Week 2: Arithmetic sequence
A school fines students 30 PHP for the first littering offense, increasing the fine by 5 PHP for each subsequent offense. The document then provides the fines for the second, third, and sixth offenses, which are 35 PHP, 40 PHP, and 55 PHP respectively. It explains that this follows an arithmetic sequence with a constant difference of 5 between terms.
Arithmetic seqence
The document defines an arithmetic sequence as a sequence where a constant difference can be added between each term to get the next term. It provides examples of finding the first term and common difference of arithmetic sequences. It explains that the common difference is found by subtracting any term from the one that follows it. Formulas are provided for finding specific terms in an arithmetic sequence given the first term, common difference, or other information. Examples problems are worked through applying these concepts.
Sequence of DM
This document provides an overview of sequences and summations in discrete mathematics. It defines a sequence as a function from a subset of natural numbers to a set, with each term of the sequence denoted as an. Examples of sequences include the terms of an arithmetic progression or geometric progression. Summations represent the sum of terms in a sequence, from an index m to n. Common summation formulas are presented, such as for arithmetic series and geometric series. The document also introduces double summations as the nested summation analog of double loops in programming.
Starr pvt. ltd. rachit's group ppt (1)
The document discusses arithmetic progressions (AP), which are sequences where the difference between consecutive terms is constant. It provides examples of AP sequences and formulas to find the nth term, the sum of terms, and other properties. Some key points:
- The nth term of an AP is given by an = a + (n-1)d, where a is the first term and d is the common difference.
- The sum of the first n terms is given by Sn = n/2 * [2a + (n-1)d].
- Examples show how to use the formulas to find individual terms, sums, number of terms, etc. given information about an AP sequence.
- Questions at
Sequence function
The document defines key concepts related to sequences and series. It explains that a sequence is an ordered list of numbers with a specific pattern or rule. A sequence function is a function whose domain is the set of natural numbers. Terms are the individual numbers in a sequence. Finite sequences have a set number of terms while infinite sequences continue without end. Partial sums refer to adding a specific number of terms. Sigma notation compactly represents the sum of terms in a sequence. The document also introduces the principle of mathematical induction as a method to prove that statements are true for all natural numbers.
Arithmetic and geometric mean
This document introduces sequences and series in mathematics. It defines a sequence as a set of numbers written in a particular order, with the n-th term written as un. A series is the sum of terms in a sequence. An arithmetic progression has terms where each new term is obtained by adding a constant difference to the preceding term. The n-th term of an arithmetic progression is a + (n - 1)d, where a is the first term and d is the common difference. A geometric progression multiplies each new term by a constant ratio r to obtain the next term, with the n-th term written as arn-1. Formulas are provided for finding the n-th term, sum of terms,
Arithmetic progression
The document discusses arithmetic progression (A.P.), which is a series where the difference between consecutive terms is constant. It provides examples like the summation of natural numbers from 1 to n. The key aspects are:
1) An A.P. has a first term, a common difference between terms, and a formula to calculate any term as the first term plus the product of the common difference and the term number minus 1.
2) The formula for summing the first n natural numbers is n(n+1)/2.
3) By analogy, the formula for summing an A.P. from its first term a to its nth term is n(2a+(n-1)d)/
Diploma_Semester-II_Advanced Mathematics_Complex number
This document provides information about complex numbers. It begins by introducing complex numbers and defining them as numbers of the form a + bi, where a and b are real numbers and i = √-1. It then discusses various representations of complex numbers including Cartesian (a + bi), polar (r(cosθ + i sinθ)), and Euler (re^iθ) forms. It also covers operations on complex numbers such as addition, subtraction, multiplication, division, and powers. De Moivre's theorem relating powers of complex numbers to trigonometric functions is presented. The document concludes by stating some basic algebraic laws for complex numbers.
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Arithmeticprogression 130714002550-phpapp02
- 2. Sequence: A list of numbers having specific relation between the consecutive terms is generally called a sequence. e.g. 1, 3, 5, 7,……… (next term to a term is obtained by adding 2 with it) & 2, 6, 18, 54,…….( next term to a term is obtained by multiplying 3 with it)
- 3. Arithmetic Progression: If various terms of a sequence are formed by adding a fixed number to the previous term or the difference between two successive terms is a fixed number, then the sequence is called AP. e.g.1) 2, 4, 6, 8, ……… the sequence of even numbers is an example of AP 2) 5, 10, 15, 20, 25….. In this each term is obtained by adding 5 to the preceding term except first term.
- 4. Illustrative example for A.P. =d,where d=1 a a+d a+2d a+3d………………
- 5. The general form of an Arithmetic Progression is a , a +d , a + 2d , a + 3d ………………, a + (n- 1)d Where ‘a’ is first term and ‘d’ is called common difference.
- 6. Common Difference - The fixed number which is obtained by subtracting any term of AP from its previous term. If we take First term of an AP as a and Common Difference as d, Then, nth term of that AP will be An = a + (n-1)d
- 7. 3, 7, 11, 15, 19 … Notice in this sequence that if we find the difference between any term and the term before it we always get 4. 4 is then called the common difference and is denoted with the letter d. d =4 To get to the next term in the sequence we would add 4 so a recursive formula for this sequence is: 41 += −nn aa The first term in the sequence would be a1 which is sometimes just written as a. a =3
- 8. 3, 7, 11, 15, 19 … +4 +4 +4 +4 Each time you want another term in the sequence you’d add d. This would mean the second term was the first term plus d. The third term is the first term plus d plus d (added twice). The fourth term is the first term plus d plus d plus d (added three times). So you can see to get the nth term we’d take the first term and add d (n - 1) times. d =4 ( )dnaan 1−+= Try this to get the 5th term. a =3 ( ) 1916341535 =+=−+=a
- 9. Let’s see an example!! Let a=2, d=2, n=12,find An An=a+(n-1)d =2+(12-1)2 =2+(11)2 =2+22 Therefore, An=24 Hence solved.
- 10. To check that a given term is in A.P. or not. 2, 6, 10, 14…. Here first term a = 2, find differences in the next terms a2-a1 = 6 – 2 = 4 a3-a2 = 10 –6 = 4 a4-a3 = 14 – 10 = 4 Since the differences are common. Hence the given terms are in A.P.
- 11. Problem : Find the value of k for which the given series is in A.P. 4, k –1 , 12 Solution : Given A.P. is 4, k –1 , 12….. If series is A.P. then the differences will be common. d1 = d1 a2 – a1 = a3 – a2 k – 1 – 4 = 12 – (k – 1) k – 5 = 12 – k + 1 k + k = 12 + 1 + 5
- 12. The sum of n terms, we find as, Sum = n X [(first term + last term) / 2] Now last term will be = a + (n-1) d Therefore, Sum(Sn ) =n X [{a + a + (n-1) d } /2 ] = n/2 [ 2a + (n+1)d]
- 13. DERIVATION The sum to n terms is given by: Sn = a + (a + d) + (a + 2d) + … + (a + (n – 1)d) (1) If we write this out backwards, we get: Sn = (a + (n – 1)d) + (a + (n – 2)d) + … +a (2) Now let’s add (1) and (2): 2Sn = [2a + (n – 1)d] + [2a + (n – 1)d] + … ……… + [2a + (n – 1)d] So, S = n/2 [2a + (n – 1)d]
- 14. Problem . Find number of terms of A.P. 100, 105, 110, 115,,……………… 500Solution. First term is a = 100 , an = 500 Common difference is d = 105 -100 = 5 nth term is an = a + (n-1)d 500 = 100 + (n-1)5 500 - 100 = 5(n – 1) 400 = 5(n – 1) 5(n – 1) = 400
- 15. 5(n – 1) = 400 n – 1 = 400/5 n - 1 = 80 n = 80 + 1 n = 81 Hence the no. of terms are 81.
- 16. Problem . Find the sum of 30 terms of given A.P. ,12 , 20 , 28 , 36……… Solution : Given A.P. is 12 , 20, 28 , 36 Its first term is a = 12 Common difference is d = 20 – 12 = 8 The sum to n terms of an arithmetic progression Sn = n/2 [ 2a + (n - 1)d ] = ½ x 30 [ 2x 12 + (30-1)x 8] = 15 [ 24 + 29 x8]
- 17. = 15[24 + 232] = 15 x 246 = 3690 THE SUM OF TERMS IS 3690
- 18. Problem . Find the sum of terms in given A.P. 2 , 4 , 6 , 8 , ……………… 200 Solution: Its first term is a = 2 Common difference is d = 4 – 2 = 2 nth term is an = a + (n-1)d 200 = 2 + (n-1)2 200 - 2 = 2(n – 1) 2(n – 1) = 198 n – 1 = 99, n = 100
- 19. The sum to n terms of an arithmetic progression Sn = n/2[ 2a + (n - 1)d ] S100 = 100/2 [ 2x 2 + (100-1)x 2] = 50 [ 4 + 198] = 50[202] = 10100
- 20. The difference between two terms of an AP can be formulated as below:- nth term – kth term = t(n) – t(k) = {a + (n-1)d} – { a + (k-1) d } = a + nd – d – a – kd + d = nd – kd Hence, t(n) – t(k) = (n – k) d