Precalculus (10th Edition)

Published by Pearson
ISBN 10: 0-32197-907-9
ISBN 13: 978-0-32197-907-0

Chapter 12 - Sequences; Induction; the Binomial Theorem - 12.3 Geometric Sequences; Geometric Series - 12.3 Assess Your Understanding - Page 825: 78


Geometric Sum: $\approx 350,319.62$

Work Step by Step

We are given the sequence: $\left\{\left(\dfrac{5}{4}\right)^n\right\}$ Compute the ratio between two consecutive terms: $\dfrac{a_{k+1}}{a_k}=\dfrac{\left(\dfrac{5}{4}\right)^{k+1}}{\left(\dfrac{5}{4}\right)^k}=\dfrac{5}{4}$ As the ratio between any consecutive terms is constant, the sequence is geometric. Its elements are: $a_1=\left(\dfrac{5}{4}\right)^1=\dfrac{5}{4}$ $r=\dfrac{5}{4}$ We determine the sum of the first 50 terms: $S_n=a_1\cdot\dfrac{1-r^n}{1-r}$ $S_{50}=\dfrac{5}{4}\cdot\dfrac{1-\left(\dfrac{5}{4}\right)^{50}}{1-\dfrac{5}{4}}\approx 350,319.62$
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