Precalculus: Concepts Through Functions, A Unit Circle Approach to Trigonometry (3rd Edition)

Published by Pearson
ISBN 10: 0-32193-104-1
ISBN 13: 978-0-32193-104-7

Chapter 11 - Sequences; Induction; the Binomial Theorem - Section 11.3 Geometric Sequences; Geometric Series - 11.3 Assess Your Understanding - Page 846: 95


A total of $1.845\times 10^{19}\text{ grains}$ are needed.

Work Step by Step

The sum $S_{n}$ of the first $n$ terms can be computed as: $S_{n}=\displaystyle \sum_{k=1}^{n}a_{1}r^{k-1}=a_{1}\cdot\frac{1-r^{n}}{1-r}$ Since, $a_{1}=1\\ a_{2}=1\cdot 2\\ a_{3}=1\cdot 2^{2}\\ a_{4}=1\cdot 2^{3}....$. This shows a geometric sequence with $a_{1}=1; r=2; n=64$ Now, the sum $S_{n}$ of the first $64$ terms can be computed as: $$S_{64}=1\cdot \displaystyle \frac{1-2^{64}}{1-2}=\frac{1-2^{64}}{-1}=2^{64}-1\approx 1.845\times 10^{19}\ \text{ grains}$$
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