College Algebra 7th Edition

Published by Brooks Cole
ISBN 10: 1305115546
ISBN 13: 978-1-30511-554-5

Chapter 8, Sequences and Series - Section 8.2 - Arithmetic Sequences - 8.2 Exercises: 60

Answer

$S_{20}=1020$

Work Step by Step

RECALL: (1) The sum of the first $n$ terms of an arithmetic sequence is given by the formula: $S_n=\dfrac{n}{2}(a+a_n)$ where $a$ = first term $d$ = common difference $a_n$ = $n^{th}$ term (2) The $n^{th}$ term $a_n$ of an arithmetic sequence is given by the formula: $a_n = a + (n-1)d$ where $a$ = first term $d$ = common difference The given arithmetic sequence has: $a=89 \\a_n = 13 \\d=85-89=-4$ The formula for the partial sum requires the values of $a$, $a_n$ and $n$. However, only $a$ and $a_n$ are known at the moment. Solve for $n$ using the formula for $a_n$ to obtain: $\require{cancel} a_n = a + (n-1)d \\13 = 89+(n-1)(-4) \\13-89 = (n-1)(-4) \\-76=(n-1)(-4) \dfrac{-76}{-4}=\dfrac{(n-1)(-4)}{-4} \\19=n-1 \\19+1=n-1+1 \\20=n$ Now that it is known that $n=20$, the sum of the first 20 terms can be computed using the formula above. $S_{20} = \dfrac{20}{2}(89+13) \\S_{20}=10(102) \\S_{20}=1020$
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