Answer
$\$98,888$ more will be from the lump-sum investment than from the annuity.
Work Step by Step
First condition Lump-Sum deposit.
$P=\$40,000$
Rate of interest compounded annually $r=6.5\%=0.065$.
Time $t=25$ years.
The compounded interest formula is $A=P(1+r)^t$.
Substitute all values into the formula.
$A=40,000(1+0.065)^{25}$
$A=193107.964325$
Round to the nearest dollar.
$A=\$193108$
Second condition Periodic deposits.
First year $=\$1600$
Second year $=\$1600(1.065)$
Third year $=\$1600(1.065)^2$ and so on.
This is the geometric sequence.
The sum of the all $25$ years deposits is
$S_{25}=\frac{1,600(1-(1.065)^{25})}{1-1.065}$
Simplify.
$S_{25}=\$94220.2857387$
Round to the nearest dollar.
$A=\$94220$.
The difference of both condition is
$\$193108-\$94220=\$98,888$.
