Algebra 2 Common Core

Published by Prentice Hall
ISBN 10: 0133186024
ISBN 13: 978-0-13318-602-4

Chapter 9 - Sequences and Series - 9-4 Arithmetic Series - Practice and Problem-Solving Exercises - Page 591: 20

Answer

$\sum\limits_{i=1}^{23} -8n + 113$

Work Step by Step

We want to find the explicit formula for the $nth$ term, so use the explicit formula for an arithmetic sequence, which is given by: $a_{n} = a_{1} + (n - 1)d$, where $a_{n}$ is a certain term in a sequence, $ a_{1}$ is the first term in the sequence, $n$ is the number of the term within the sequence, and $d$ is the common difference. The common difference $d$ is found by subtracting one term in the sequence from the one directly following it: $d = 97 - 105 = -8$ The common difference $d$ is $-8$. To find the explicit formula for the $nth$ term, plug in the values we know into the explicit formula for an arithmetic sequence: $a_{n} = 105 + (n - 1)(-8)$ Use distributive property: $a_{n} = 105 + (n - 1)(-8)$ Combine like factors on the right side of the equation: $a_{n} = 105 - 8n + 8$ Combine like terms: $a_{n} = -8n + 113$ Now that we have the explicit formula to find the value of $n$, find the value of $n$ for the last term $-71$ by plugging in $-71$ into the explicit formula: $-71 = -8n + 113$ Subtract $113$ from each side of the equation: $-184 = -8n$ Divide each side of the equation by $-8$: $n = 23$ This means that $-71$ is the $23rd$ and final term in this series. Now, we can plug our values into the summation notation, where we plug in the explicit formula for finding the $nth$ term with the lower limit being $1$ and the upper limit being the value of $n$ for the last term in the series: $\sum\limits_{i=1}^{23} -8n + 113$
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