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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. 295 (page 1219)

In the following exercises, find the sum.
${鈭�}_{i = 1}^{6} 24 {(\frac{1}{2})}^{i}$

Short Answer

Expert verified

The sum is $23.625$

Step by step solution

01

Step 1. Given information

The given summation notation is ${鈭�}_{i = 1}^{6} 24 {(\frac{1}{2})}^{i}$

02

Step 2. Expand the summation notation.

${鈭�}_{i = 1}^{6} 24 {(\frac{1}{2})}^{i} = 24 {(\frac{1}{2})}^{1} + 24 {(\frac{1}{2})}^{2} + 24 {(\frac{1}{2})}^{3} + . . . + 24 {(\frac{1}{2})}^{6} = 12 + 6 + 3 + . . . + \frac{3}{8}$

Now this sequence is in G.P. with common ratio $= \frac{1}{2}$ .

03

Step 3. Find the sum.

$S_{n} = \frac{a_{1} (1 - r^{n})}{1 - r}$

$a_{1} = 12 r = \frac{1}{2} n = 6$

So the sum is

$S_{6} = \frac{12 (1 - {(\frac{1}{2})}^{6})}{1 - \frac{1}{2}} S_{6} = 24 (1 - {(\frac{1}{2})}^{6}) S_{6} = 23.625$

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Most popular questions from this chapter

In the following exercises, write the first five terms of each arithmetic sequence with the given first term and common difference.

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