Answer
Geometric;
Sum: $\dfrac{125}{3}\left(1-\left(\dfrac{2}{5}\right)^n\right)$
Work Step by Step
We are given the sequence:
$25,10,4,\dfrac{8}{5},....$
Determine the ratio of consecutive terms:
$\dfrac{a_2}{a_1}=\dfrac{10}{25}=\dfrac{2}{5}$
$\dfrac{a_3}{a_2}=\dfrac{4}{10}=\dfrac{2}{5}$
$\dfrac{a_4}{a_3}=\dfrac{\dfrac{8}{5}}{4}=\dfrac{2}{5}$
As the ratio of consecutive terms is constant, the sequence is geometric. Its elements are:
$a_1=25$
$d=\dfrac{2}{5}$
Determine the sum of the first $n$ terms:
$S_n=a_1\cdot\dfrac{1-r^n}{1-r}$
$S_n=25\cdot\dfrac{1-\left(\dfrac{2}{5}\right)^n}{1-\dfrac{2}{5}}=25\cdot \left(1-\left(\dfrac{2}{5}\right)^n\right)\cdot\dfrac{5}{3}$
$=\dfrac{125}{3}\left(1-\left(\dfrac{2}{5}\right)^n\right)$
