#### Answer

(a) We will have $\$98,888$ more from the lump-sum investment than from the annuity.
(b) The lump-sum investment earns $\$98,888$ more in interest than the annuity.

#### Work Step by Step

(a) This is the formula we use when we make calculations with compound interest:
$A = P~(1+\frac{r}{n})^{nt}$
$A$ is the final amount in the account
$P$ is the principal (the amount of money invested)
$r$ is the interest rate
$n$ is the number of times per year the interest is compounded
$t$ is the number of years
We can find the total amount in the account at the end of 25 years when we invest a lump sum at a rate of 6.5% compounded annually.
$A = P~(1+\frac{r}{n})^{nt}$
$A = (\$40,000)~(1+\frac{0.065}{1})^{(1)(25)}$
$A = \$193,108$
After 25 years, there will be $\$193,108$ in the account.
This is the formula we use to calculate the value of an annuity:
$A = \frac{P~[(1+\frac{r}{n})^{nt}-1]}{\frac{r}{n}}$
$A$ is the future value of the annuity
$P$ is the amount of the periodic deposit
$r$ is the interest rate
$n$ is the number of times per year the interest is compounded
$t$ is the number of years
$A = \frac{P~[(1+\frac{r}{n})^{nt}~-1]}{\frac{r}{n}}$
$A = \frac{(\$1600)~[(1+\frac{0.065}{1})^{(1)(25)}~-1]}{\frac{0.065}{1}}$
$A = \$94,220$
The value of the annuity is $\$94,220$
We can calculate the difference between the lump-sum investment and the value of the annuity.
$\$193,108 - \$94,220 = \$98,888$
We will have $\$98,888$ more from the lump-sum investment than from the annuity.
The lump-sum investment earns $\$98,888$ more in interest than the annuity.