Intermediate Algebra for College Students (7th Edition)

Published by Pearson
ISBN 10: 0-13417-894-7
ISBN 13: 978-0-13417-894-3

Chapter 11 - Section 11.2 - Arithmetic Sequences - Exercise Set: 36

Answer

$S_{25}=3,775$

Work Step by Step

RECALL: (1) The sum of the first $n$ terms, $s_n$, of an arithmetic sequence can be found using the formula $S_n = \frac{n}{2}(a_1+a_n)$ where $a_1$=first term $a_n$ = $n^{th}$ term (2) The $n^{th}$ term, $a_n$, of an arithmetic sequence can be found using the formula $a_n=a_1 + d(n-1)$ where $d$=common difference $a_1$ = first term To find the sum of the first 25 terms of the sequence, we need to find the value of $a_{25}$. However, the value of $a_{25}$ can only be found if we know the value of $d$. The terms of the sequence increase by 12 so the common difference is $d=12$. The first term of the sequence is $7$ so $a_1=7$. Substitute these values into the formula in (2) above to obtain; $a_n= 7+12(n-1)$ Solve for the 25th term of the sequence to obtain: $a_{25} = 7 + 12(25-1) \\a_{25} = 7 + 12(24) \\a_{25} = 7 + 288 \\a_{25} = 295$ Solve for the sum of the first 25 terms using the formula in (1) above to obtain: $S_{25} = \frac{25}{2}(7+295) \\S_{25} = \frac{25}{2}(302) \\S_{25} = 3775$
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