Given:
First 10 term of a geometric series are:
[tex]\dfrac{16}{25}+\dfrac{8}{5}+4+10+...+a_10[/tex]
To find:
The expression that represents the sum of the series.
Solution:
The first term of the given geometric series is:
[tex]a_1=\dfrac{16}{25}[/tex]
The common ratio of the given geometric series is:
[tex]r=\dfrac{a_2}{a_1}[/tex]
[tex]r=\dfrac{\dfrac{8}{5}}{\dfrac{16}{25}}[/tex]
[tex]r=\dfrac{8}{5}\times \dfrac{25}{16}[/tex]
[tex]r=\dfrac{5}{2}[/tex]
The sum of first n terms of a geometric sequence is:
[tex]S_n=\dfrac{a_1(1-r^n}{1-r}[/tex]
Where r is the common ratio.
Putting [tex]a_1=\dfrac{16}{25},r=\dfrac{5}{2},n=10[/tex] in the above formula, we get
[tex]S_{10}=\dfrac{16}{25}\left(\dfrac{(1-(\dfrac{5}{2})^{10}}{1-\dfrac{5}{2}}\right)[/tex]
Therefore, the correct option is C, i.e., [tex]\dfrac{16}{25}\left(\dfrac{(1-(\dfrac{5}{2})^{10}}{1-\dfrac{5}{2}}\right)[/tex].
