Answer
$\dfrac{121}{3}$
Work Step by Step
We calculate the given sum by expanding it: we calculate each term for $m=1,2,3,4,5$ and add them:
$$\begin{align*}
\sum_{m=1}^5 3^{m-2}&=3^{-1}+3^0+3^1+3^2+3^3\\
&=\dfrac{1}{3}+1+3+9+27\\
&=\dfrac{121}{3}.
\end{align*}$$
