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Intermediate Algebra

OPENSTAX

$Math Studyset 91影视 Explanations$ Math

OER 2017 Edition 路 1346 pages

ISBN: 9780998625720

Chapter 12: Q 168. (page 1199)

In the following exercises, find the sum of the geometric sequence.
${鈭�}_{i = 1}^{15} 4 {(\frac{1}{2})}^{i}$

Short Answer

Expert verified

The obtained sum is 4.

Step by step solution

01

Step 1. Given Information.

Consider the given geometric sequence.

${鈭�}_{i = 1}^{15} 4 {(\frac{1}{2})}^{i}$

02

Step 2. Express in terms of the given sequence.

Write the few terms of the sequence.

$\begin{array}{rcl} {鈭�}_{i = 1}^{15} 4 {(\frac{1}{2})}^{i} & = & 4 {(\frac{1}{2})}^{1} + 4 {(\frac{1}{2})}^{2} + 4 {(\frac{1}{2})}^{3} + 鈰� \\ = & 2 + 1 + \frac{1}{2} + 鈰� \end{array}$

03

Step 3. Find the common ratio.

The obtained sequence is ${鈭�}_{i = 1}^{15} 4 {(\frac{1}{2})}^{i} = 2 + 1 + \frac{1}{2} + 鈰�$ .

The first term is 2. Now find the common ratio.

$r = \frac{1}{2}, \frac{\frac{1}{2}}{1} = \frac{1}{2}$

Thus, the common ratio is $\frac{1}{2}$ .

04

Step 4. Find the sum of the given sequence.

The sum formula of the geometric sequence is $S_{n} = \frac{a_{1} (1 - r^{n})}{1 - r}$ .

Substitute the value of the first and the common ratio in the sum formula.

$\begin{array}{rcl} S_{15} & = & \frac{2 (1 - {(\frac{1}{2})}^{15})}{1 - \frac{1}{2}} \\ = & \frac{2 (1 - {(\frac{1}{2})}^{15})}{\frac{2 - 1}{2}} \\ = & 4 (1 - {(\frac{1}{2})}^{15}) \\ = & 4 - {(\frac{1}{2})}^{13} \\ 鈮� & 4 \end{array}$

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