An annuity in perpetuity is one that continues forever. Such annuities are useful in setting up scholarship funds to ensure that the award continues.
a. Draw a time line (as in Example 1 ) to show that to set up an annuity in perpetuity of amount
$R$ per time period, the amount that must be invested now is
$$
A_{p}=\frac{R}{1+i}+\frac{R}{(1+i)^{2}}+\frac{R}{(1+i)^{3}}+\cdots+\frac{R}{(1+i)^{n}}+\cdots
$$
where $i$ is the interest rate per time period.
b. Find the sum of the infinite series in part (a) to show that
$$
A_{p}=\frac{R}{i}
$$
c. How much money must be invested now at $10 \%$ per year, compounded annually, to provide an annuity in perpetuity of $\$ 5000$ per year? The first payment is due in 1 year.
d. How much money must be invested now at $8 \%$ per year, compounded quarterly, to provide an annuity in perpetuity of $\$ 3000$ per year? The first payment is due in 1 year.