#### Answer

(a) We will have $\$30,000$ more from the lump-sum investment than from the annuity.
(b) The lump-sum investment earns $\$30,000$ more 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 20 years when we invest a lump sum at a rate of 5% compounded annually.
$A = P~(1+\frac{r}{n})^{nt}$
$A = (\$30,000)~(1+\frac{0.05}{1})^{(1)(20)}$
$A = \$79,599$
After 20 years, there will be $\$79,599$ 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
$A = \frac{P~[(1+\frac{r}{n})^{nt}~-1]}{\frac{r}{n}}$
$A = \frac{(\$1500)~[(1+\frac{0.05}{1})^{(1)(20)}~-1]}{\frac{0.05}{1}}$
$A = \$49,599$
The value of the annuity is $\$49,599$
We can calculate the difference between the lump-sum investment and the value of the annuity.
$\$79,599 - \$49,599 = \$30,000$
We will have $\$30,000$ more from the lump-sum investment than from the annuity.