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: 15

Answer

$\sum\limits_{i=1}^{5} 4n$

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: $d = 8 - 4 = 4$ The common difference $d$ is $4$. To find the explicit formula for the $nth$ term, plug in the values we know into the explicit formula for an arithmetic sequence: $a_{n} = 4 + (n - 1)(4)$ Use distributive property: $a_{n} = 4 + 4n - 4$ Combine like factors on the right side of the equation: $a_{n} = 4n$ Now that we have the explicit formula to find the value of $n$, find the value of $n$ for the last term $20$ by plugging in $20$ into the explicit formula: $20 = 4n$ Divide each side of the equation by $4$: $n = 5$ This means that $20$ is the $5th$ and final term in this series. Now, we can plug our values into the summation notation: $\sum\limits_{i=1}^{5} 4n$
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