Chapter 12: Q. 4 (page 839)

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

Short Answer

Expert verified

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

Step by step solution

Given information:

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

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

To expand the given sum:

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

Simplification:

= [\frac{2}{3} - 1] + [\frac{4}{9} - 2] + [\frac{8}{27} - 3] + [\frac{16}{81} - 4] = [\frac{2 - 3}{3}] + [\frac{4 - (2 \times 9)}{9}] + [\frac{8 - (3 \times 27)}{27}] + [\frac{16 - (4 \times 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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Write out the sum. Evaluate the sum.
$\sum_{k = 1}^{4} [{(\frac{2}{3})}^{k} - k]$

Short Answer

Step by step solution

Given information:

To expand the given sum:

Simplification:

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Write out the sum. Evaluate the sum.∑k=1423k-k

Short Answer

Step by step solution

Given information:

To expand the given sum:

Simplification:

One App. One Place for Learning.

Most popular questions from this chapter

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