Question

Write out the sum. Evaluate the sum.

$\sum _{k=1}^4\left[{\left(\frac{2}{3}\right)}^k-k\right]$

Short answer

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

Step-by-step solution

Given information:

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

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

To expand the given sum:

$\sum _{k=1}^4\left[{\left(\frac{2}{3}\right)}^k-k\right]=\left[{\left(\frac{2}{3}\right)}^1-1\right]+\left[{\left(\frac{2}{3}\right)}^2-2\right]+\left[{\left(\frac{2}{3}\right)}^3-3\right]+\left[{\left(\frac{2}{3}\right)}^4-4\right]$

Simplification:

$$\begin{gathered} =\left[\frac{2}{3}-1\right]+\left[\frac{4}{9}-2\right]+\left[\frac{8}{27}-3\right]+\left[\frac{16}{81}-4\right] \\ =\left[\frac{2-3}{3}\right]+\left[\frac{4-\left(2\times 9\right)}{9}\right]+\left[\frac{8-(3\times 27)}{27}\right]+\left[\frac{16-(4\times 81)}{81}\right] \\ =\left[\frac{-1}{3}\right]+\left[\frac{-14}{9}\right]+\left[\frac{-73}{27}\right]+\left[\frac{-308}{81}\right] \\ =\frac{-27-126-219-308}{81} \\ =\frac{-680}{81} \end{gathered}$$