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Intermediate Algebra

OPENSTAX

$Math Studyset Vaia Explanations$ Math

OER 2017 Edition · 1346 pages

ISBN: 9780998625720

Chapter 12: Q 126. (page 1185)

In your own words, explain how to find the sum of the first n terms of an arithmetic sequence without adding all the terms.

Short Answer

Expert verified

By using the arithmetic sequence sum formula, we can find the nth term sum.

Step by step solution

01

Step 1. Introduction.

If we have a sequence of nth terms. By substituting the first term and nth term value in the sum formula to find the nth term sum.

02

Step 2. Explanation.

Consider the sum formula of the nth term.

$S_{n} = \frac{n}{2} (a_{1} + a_{n})$

Now, substitute the value of a_n as $a_{n} = a_{1} + (n - 1) d$ .

$\begin{array}{rcl} S_{n} & = & \frac{n}{2} (a_{1} + a_{n}) \\ = & \frac{n}{2} (a_{1} + a_{1} + (n - 1) d) \\ = & \frac{n}{2} (2 a_{1} + (n - 1) d) \end{array}$

Let's supposed that the first term is 4 and the value of d is 2.

$\begin{array}{rcl} S_{n} & = & \frac{n}{2} (2 (4) + (n - 1) 2) \\ = & n (2 + n - 1) \\ = & n^{2} + n \end{array}$

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Most popular questions from this chapter

Find the eleventh term of a sequence where the ninth term is 8 and the common difference is −3. Give the formula for the general term.

Find the sum of the first 50 terms of the arithmetic sequence whose general term is $a_{n} = 2 n - 5$ .

In the following exercises, determine if each sequence is arithmetic, geometric or neither. If arithmetic, indicate the common difference. If geometric, indicate the common ratio.

$4, 8, 12, 24, 48, 96, . . .$

In the following exercises, expand each binomial using Pascal's Triangle.

$(x - y)^{5}$

Find the twenty-seventh term of a sequence where the first term is 7 and the common difference is 9.

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