$2.(24$ pts$)$ Suppose $S={\displaystyle \underset{n=1}{\overset{}{}}}{a}_n,s_n$ is the $n^{th}$ partial sum of $S,$ and the terms of the series are defined by $a_n=\frac{5(3{)}^{n+1}}{4^n}.($a$.$ and b$.$ are $5$ points each, c$.$ and d$.$ are $7$ points each$)$

a$.$ Write out the first $3$ terms of the sequence $a_n.$

$a_1=$

: $=$

$a_3=$

b$.$ Write out the first $3$ terms of the sequence $s_n($Show the terms and the value of the partial sum$).$

$s_1=,$:$=$

c$.$ Determine if the sequence $a_n$ converges or diverges. If it converges, find the limit$.$ Justify your answer.

d$.$ Determine if the series $S$ converges or diverges. If it converges, find the limit$.$ Justify your answer.