Evaluate sigma-summation Underscript lower i equals 1 overscript infinity EndScripts 26 left parenthesis start fraction 1 over 5 end fraction right parenthesis superscript lower i baseline.

Question

(1 point)
Responses

26
26

6.5
6.5

32.5
32.5

130

Answer 1

To evaluate the summation

$$\sum _{i=1}^{\mathrm{\infty }}26{\left(\frac{1}{5}\right)}^i,$$

we recognize this as a geometric series of the form

$$\sum _{i=0}^{\mathrm{\infty }}a{r}^i,$$

where $a$ is the first term and $r$ is the common ratio. In our case:

• The first term $a$ for $i=1$ is $26\frac{1}{5}=\frac{26}{5}$,
• The common ratio $r=\frac{1}{5}$.

The formula for the sum of an infinite geometric series is given by:

$$S=\frac{a}{1-r},$$

provided that $|r|<1$. Here, $|r|=\frac{1}{5}<1$, so we can use the formula:

Calculating the sum:

$$S=\frac{\frac{26}{5}}{1-\frac{1}{5}}=\frac{\frac{26}{5}}{\frac{4}{5}}=\frac{26}{5}\cdot \frac{5}{4}=\frac{26}{4}=6.5.$$

Thus, the value of the summation is

$$\sum _{i=1}^{\mathrm{\infty }}26{\left(\frac{1}{5}\right)}^i=6.5.$$

The correct response is:

6.5