Thinking Mathematically (6th Edition)

Published by Pearson
ISBN 10: 0321867327
ISBN 13: 978-0-32186-732-2

Chapter 5 - Number Theory and the Real Number System - 5.7 Arithmetic and Geometric Sequences - Exercise Set 5.7 - Page 330: 119


The sequence is geometric and the sum of its first ten terms is $-1023$.

Work Step by Step

The terms have a common ratio of $-2$ so the sequence is geometric with $r=-2$ and $a_1=3$. The sum $S_n$ of the first $n$ terms of a geometric sequence is given by the formula: $$S_n=\frac{a_1(1-r^n)}{1-r}$$ The given sequence has $a_1=3$ and $r=-2$. However, the value of $a_{10}$ (the tenth term) is not yet known. Solve for $a_{10}$ using the formula $a_n=a_1\cdot r^{n-1}$ to obtain: \begin{align*} a_{10}&=3 \cdot (-2)^{10-1}\\ &=3\cdot (-2)^9\\ &=3\cdot (-512)\\ &=-1536\end{align*} Use the formula for the sum of the first $n$ terms to find the sum of the first $10$ terms (use $n=10$) to obtain: \begin{align*} S_{10}&=\frac{a_1(1-r^n)}{1-r}\\\\ &=\frac{3(1-(-2)^{10})}{1-(-2)}\\\\ &=\frac{3(1-1024)}{1+2}\\\\ &=\frac{3(-1023)}{3}\\\\ &=-1023 \end{align*}
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