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: 120

Answer

The sequence is geometric and the sum of its first ten terms is $-59,048$.

Work Step by Step

The terms have a common ratio of $-3$ so the sequence is geometric with $r=-3$ and $a_1=4$. 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 $a_1=4$ and $r=-3$. 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}&=4 \cdot (-3)^{10-1}\\ &=4\cdot (-3)^9\\ &=4(-19683)\\ &=-78,732\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{4(1-(-3)^{10})}{1-(-3)}\\\\ &=\frac{4(1-59049}{1+3}\\\\ &=\frac{4(-59048)}{4}\\\\ &=-59048 \end{align*}
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