Answer
$\$30000$.
Work Step by Step
For the periodic sum deposit, we calculate the annuity by:
$A=\frac{P\left[ {{\left( 1+\frac{r}{n} \right)}^{nt}}-1 \right]}{\left( \frac{r}{n} \right)}$
From the given information, $P=\$1500$ and $r=0.05$ for $t=20\,\text{years}$. It is compounded annually, hence $n=1$.
Thus:
$\begin{align}
& A=\frac{1500\left[ {{\left( 1+\frac{0.05}{1} \right)}^{20}}-1 \right]}{\left( \frac{0.05}{1} \right)} \\
& =\frac{1500\left[ {{\left( 1.05 \right)}^{20}}-1 \right]}{\left( 0.05 \right)} \\
& =\frac{1500\left[ \left( 2.653 \right)-1 \right]}{\left( 0.05 \right)} \\
& =49590
\end{align}$
For the lump sum deposit, we need to calculate the amount that is given by;
$A=P\left[ {{\left( 1+\frac{r}{n} \right)}^{nt}} \right]$
For the lump sum deposit $P=\$30000$ and $r=5\%$ and it is compounded annually.
Thus:
$\begin{align}
& A=30000\left[ {{\left( 1+\frac{0.05}{1} \right)}^{20}} \right] \\
& =30000\left[ 2.653 \right] \\
& =79590
\end{align}$
The difference between the lump sum and periodic deposit is:
$amount-annuity=79590-49590=30000$
