Precalculus (6th Edition) Blitzer

Published by Pearson
ISBN 10: 0-13446-914-3
ISBN 13: 978-0-13446-914-0

Chapter 10 - Section 10.3 - Geoetric Sequences and Series - Exercise Set - Page 1076: 94


You can find the infinite sum as follows: ${{S}_{n}}=\frac{{{a}_{1}}}{\left( 1-r \right)}$

Work Step by Step

The general form of the geometric sequence is$a,ar,a{{r}^{2}},a{{r}^{3}},\cdots $. The formula of summation is given by: ${{S}_{n}}=\frac{{{a}_{1}}\left( 1-{{r}^{n}} \right)}{\left( 1-r \right)}$ Here $r\ne 0$ is the common ratio and $a$ is the scale factor or start value. In an infinite geometric series, we can find the sum when $\left| r \right|<1$. $\begin{align} & {{S}_{n}}=\frac{{{a}_{1}}\left( 1-{{r}^{\infty }} \right)}{\left( 1-r \right)} \\ & =\frac{{{a}_{1}}\left( 1-0 \right)}{\left( 1-r \right)} \\ & =\frac{{{a}_{1}}}{\left( 1-r \right)} \end{align}$ Thus, the formula of summation for the infinite geometric series is given by ${{S}_{n}}=\frac{{{a}_{1}}}{\left( 1-r \right)}$
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