## Precalculus: Concepts Through Functions, A Unit Circle Approach to Trigonometry (3rd Edition)

$S_n=5[1-( \dfrac{3}{5})^n]$
The sum of the first $n$ terms of a Geometric Sequence is given by: $S_{n}=\displaystyle\sum_{k=1}^n a_1r^{k-1}=a_{1} (\dfrac{1-r^{n}}{1-r}) ; \ r\neq 0,1$ We are given: $a_1=2$ and $r=\frac{3}{5}$ We can write the sum as: $S_n=\displaystyle\sum_{k=1}^n (2) (\dfrac{3}{5})^{n-1}$ Now, $S_n= (2) \ [\dfrac{1-(\dfrac{3}{5} )^{n}}{1- \dfrac{3}{5} } \ ] \\=(2) \ [\dfrac{1-(\dfrac{3}{5} )^{n}}{ \dfrac{2}{5} } \ ]$ Therefore, $S_n=5[1-( \dfrac{3}{5})^n]$