Answer
$(a)$.semi-annualy compounded,$r=2.5\%$,provides the best Investment
Work Step by Step
The formula for continuously compounded Interest is, $A(t)=Pe^{rt}$ whereas, $P=Initial-Investment$, $r=rate$, $e=natural-logarithm's-constant-base$ and $t=time$. And formula for periodically(annualy,semi-annualy,quarterly,monthly,daily and so on) compounded Interest is, $A(t)=P\left(1+\frac{r}{n} \right)^{nt}$ whereas, $P=Intial-Investment,$ $rate,$ $n=number-of-times-it-compounds,$ and $time$.
(a) is for compounded semi-annualy, (b) is for compounded monthly and (c) is for compounded continuously.
Thus, The Table is as follows:
$\begin{array}{ll}
(a).r=2.5\% & (b).r=2.25\% & (c).r=2\%\\
P(1+\frac{r}{n})^{nt} & P(1+\frac{r}{n})^{nt} & Pe^{rt}\\
P(1+\frac{0.025}{2})^{2t} & P(1+\frac{0.0225}{12})^{12t} & Pe^{0.02t}\\
P(1.0125)^{2t} & P(1.001875)^{12t} & Pe^{0.02t}\\
P(1.0252)^t & P(1.0227)^t & P(1.0202)^t
\end{array}$
$P(1.0252)^t\gt P(1.0227)^t\gt P(1.0202)^t$. Therefore, $(a).r=2.5\%$ provides the best Investment