College Algebra 7th Edition

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

Chapter 8, Sequences and Series - Chapter 8 Review - Exercises - Page 640: 33

Answer

The common ratio of the $\{a_nb_n\}$ sequence is the product of the common ratios of the two given geometric sequences.

Work Step by Step

We are given the geometric sequences: $$\begin{align*} \{a_n\}&\text{ with common ratio }r\\ \{b_n\}&\text{ with common ratio }q.\end{align*}$$ We have to study the sequence: $$c_n=a_nb_n.$$ We calculate the ratio between two consecutive terms: $$\begin{align*} \dfrac{c_{k+1}}{c_k}&=\dfrac{a_{k+1}b_{k+1}}{a_kb_k}\\ &=\dfrac{a_{k+1}}{a_k}\cdot \dfrac{b_{k+1}}{b_k}\\ &=rq. \end{align*}$$ We got the $c_{k+1}/c_k=rq$, therefore constant. It follows that the sequence $\{c_n\}$ is also a geometric sequence with common ratio equal to the product of the common ratios of the two initial geometric sequences.
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