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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. 4 (page 839)

Write out the sum. Evaluate the sum.
${鈭�}_{k = 1}^{4} [{(\frac{2}{3})}^{k} - k]$

Short Answer

Expert verified

The value of the given sum is $\frac{- 680}{81}$ .

Step by step solution

01

Given information:

Consider , ${鈭�}_{k = 1}^{4} [{(\frac{2}{3})}^{k} - k]$

To expand the sum and then to find the value of it.

02

To expand the given sum:

${鈭�}_{k = 1}^{4} [{(\frac{2}{3})}^{k} - k] = [{(\frac{2}{3})}^{1} - 1] + [{(\frac{2}{3})}^{2} - 2] + [{(\frac{2}{3})}^{3} - 3] + [{(\frac{2}{3})}^{4} - 4]$

03

Simplification:

= [\frac{2}{3} - 1] + [\frac{4}{9} - 2] + [\frac{8}{27} - 3] + [\frac{16}{81} - 4] = [\frac{2 - 3}{3}] + [\frac{4 - (2 脳 9)}{9}] + [\frac{8 - (3 脳 27)}{27}] + [\frac{16 - (4 脳 81)}{81}] = [\frac{- 1}{3}] + [\frac{- 14}{9}] + [\frac{- 73}{27}] + [\frac{- 308}{81}] = \frac{- 27 - 126 - 219 - 308}{81} = \frac{- 680}{81}

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Most popular questions from this chapter

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

${鈭�}_{k = 0}^{n} \frac{1}{3^{k}}$

In Problems 17鈥�28, write down the first five terms of each sequence.

$\{t_{n}\} = \{\frac{{(- 1)}^{n}}{(n + 1 (n + 2)}\}$

In Problems 51鈥�66, determine whether each infinite geometric series converges or diverges. If it converges, find its sum.

{鈭�}_{k = 1}^{鈭�} 8 {(\frac{1}{3})}^{k - 1}

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

${鈭�}_{k = 1}^{n} \frac{k^{2}}{2}$

In Problems 17鈥�28, write down the first five terms of each sequence.

$\{d_{n}\} = \{{(- 1)}^{n - 1} (\frac{n}{2 n - 1})\}$

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