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 332: 153


The provided statement: A geometric series \[a,ar,a{{r}^{2}},a{{r}^{3}},a{{r}^{4}},\ldots \] is True.

Work Step by Step

If a sequence is geometric, we can write as many terms as we want by repeatedly multiplying by the common ratio. It is known that the geometric sequence is; \[a,ar,a{{r}^{2}},a{{r}^{3}},a{{r}^{4}},\ldots \] Where a is first term and r is common ratio; \[\begin{align} & r=\frac{{{a}_{2}}}{{{a}_{1}}} \\ & =\frac{{{a}_{3}}}{{{a}_{2}}} \\ & =\frac{{{a}_{3}}}{{{a}_{4}}} \end{align}\] For example, if a sequence is:\[2,4,8,16,32,.\ldots \] Where common ratio,\[r=\frac{{{a}_{2}}}{{{a}_{1}}}=2\] \[\begin{align} & {{a}_{1}}=2 \\ & {{a}_{2}}=2\times 2=4 \\ & {{a}_{3}}=4\times 2=8 \\ & {{a}_{4}}=8\times 2=16 \\ \end{align}\] Thus, repeatedly multiply by the common ratio to make a geometric sequence. Hence, the given statement is true.
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