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Precalculus Enhanced with Graphing Utilities

Sullivan

$Math Studyset 91影视 Explanations$ Math

6th Edition 路 1200 pages

ISBN: 9780321795465

Chapter 12: Q 44. (page 825)

In Problems 39鈥� 44, find each sum.
$2 + \frac{6}{5} + \frac{18}{25} + . . . + 2 {(\frac{3}{5})}^{n - 1}$

Short Answer

Expert verified

The sum of the given sequence is $5 路 1 - {\frac{3}{5}}^{n}$

Step by step solution

01

Step 1. Write the information.

The given sequence is $2 + \frac{6}{5} + \frac{18}{25} + . . . + 2 {(\frac{3}{5})}^{n - 1}$

02

Step 2. Find the common ratio.

$a_{1} = 2, a_{2} = \frac{6}{5}, a_{3} = \frac{18}{25} \frac{a_{2}}{a_{1}} = \frac{\frac{6}{5}}{2} = \frac{3}{5} \frac{a_{3}}{a_{2}} = \frac{\frac{18}{25}}{\frac{6}{5}} = \frac{3}{5} r = \frac{3}{5}$

03

Step 3. Find the sum

The formula for the sum of nth term is $s_{n} = \frac{a_{1} (1 - r^{n})}{1 - r}$

$s_{n} = 2 路 \frac{1 - {\frac{3}{5}}^{n}}{1 - \frac{3}{5}} = 2 路 \frac{1 - {\frac{3}{5}}^{n}}{\frac{2}{5}} = 5 路 1 - {\frac{3}{5}}^{n}$

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Most popular questions from this chapter

In Problems 51鈥�60, write out each sum.

${鈭�}_{k = 0}^{n} \frac{1}{3^{k}}$

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In Problems 29鈥�36, the given pattern continues. Write down the nth term of a sequence $\{a_{n}\}$ suggested by the pattern.

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