Evaluate the series ${\displaystyle \underset{k=1}{\overset{}{}}}(\frac{8}{5^k}-\frac{8}{5^{k+1}})$ two ways as outlined in parts $($a$)$ and $($b$).$

a$.$ Evaluate ${\displaystyle \underset{k=1}{\overset{}{}}}(\frac{8}{5^k}-\frac{8}{5^{k+1}})$ using a telescoping series argument.

b$.$ Evaluate ${\displaystyle \underset{k=1}{\overset{}{}}}(\frac{8}{5^k}-\frac{8}{5^{k+1}})$ using a geometric series argument after first simplifying $\frac{8}{5^k}-\frac{8}{5^{k+1}}$ by obtaining a common denominator

a$.$ Write the nth term of the partial sum.

$S_n=(\frac{8}{5^1}-\frac{8}{5^2})+(\frac{8}{5^2}-\frac{8}{5^3})+(\frac{8}{5^3}-\frac{8}{5^4})+$cdots$+$

$($Type an exact answer.$)$