Question

The first 4 terms of a 10-term geometric series are listed below. Which expression represents the sum

Answer 1

Given:

First 10 term of a geometric series are:

$\dfrac{16}{25}+\dfrac{8}{5}+4+10+...+a_10$

To find:

The expression that  represents the sum of the series.

Solution:

The first term of the given geometric series is:

$a_1=\dfrac{16}{25}$

The common ratio of the given geometric series is:

$r=\dfrac{a_2}{a_1}$

$r=\dfrac{\dfrac{8}{5}}{\dfrac{16}{25}}$

$r=\dfrac{8}{5}\times \dfrac{25}{16}$

$r=\dfrac{5}{2}$

The sum of first n terms of a geometric sequence is:

$S_n=\dfrac{a_1(1-r^n}{1-r}$

Where r is the common ratio.

Putting $a_1=\dfrac{16}{25},r=\dfrac{5}{2},n=10$ in the above formula, we get

$S_{10}=\dfrac{16}{25}\left(\dfrac{(1-(\dfrac{5}{2})^{10}}{1-\dfrac{5}{2}}\right)$

Therefore, the correct option is C, i.e., $\dfrac{16}{25}\left(\dfrac{(1-(\dfrac{5}{2})^{10}}{1-\dfrac{5}{2}}\right)$.