Explanation The arithmetic sequence with the first term a and the common difference d is the sequence defined by a 1 = a and a k + 1 = a k + d for k ≥ 1 . We have to first show that the formula for the n th term of an arithmetic sequence is a n = a + ( n − 1 ) d . We will prove this by the principle of mathematical induction. For n = 1 , a 1 = a + ( 1 − 1 ) d = a which is true. Let us assume that the statement is true for n = k . Thus, a k = a + ( k − 1 ) d . We will now prove that the statement is true for n = k + 1 . That is, a k + 1 = a + ( k + 1 − 1 ) d = a + k d . Now, by the definition of arithmetic sequence a k + 1 = a k + d . Using the induction hypothesis, a k + 1 = a + ( k − 1 ) d + d = a + k d − d + d = a + k d Thus, the result is true for n = k + 1 . Hence, we conclude that the n th term of an arithmetic sequence is a n = a + ( n − 1 ) d for n ≥ 1 . Now, we have to first show that the formula for the sum of the first n terms of the sequence is S n = n 2 [ 2 a + ( n − 1 ) d ] . We will prove this by the principle of mathematical induction. For n = 1 , S 1 = 1 2 [ 2 a + ( 1 − 1 ) d ] = 1 2 [ 2 a ] = a which is true. Let us assume that the statement is true for n = k . Thus, S k = k 2 [ 2 a + ( k − 1 ) d ] . We will now prove that the statement is true for n = k + 1 . That is, S k + 1 = k + 1 2 [ 2 a + ( k + 1 − 1 ) d ] = k + 1 2 [ 2 a + k d ] Now, by the definition of arithmetic sequence a k + 1 = a k + d

3rd Edition

ISBN: 9780131679955

Author: Edgar G. Goodaire

Publisher: Prentice Hall

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Sequence and Series

A sequence is defined as a collection of things. Series is defined to sum the things one by one in the sequence. It was invented by German mathematician Carl Friedrich Gauss.

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Chapter 5.2, Problem 22E

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The formulas (6) and (7) for the nth term and the sum of the first n terms of the arithmetic sequence with first term a and common difference d.

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Chapter 5.2, Problem 21E

Chapter 5.2, Problem 23E

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Discrete Mathematics with Graph Theory

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The formulas (6) and (7) for the n th term and the sum of the first n terms of the arithmetic sequence with first term a and common difference d .

Solution Summary: The author explains the formulas (6) and (7) for the n th term and the sum of the first n terms of an arithmetic sequence.

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The formulas (6) and (7) for the nth term and the sum of the first n terms of the arithmetic sequence with first term a and common difference d.

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