## Thinking Mathematically (6th Edition)

(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.
(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.