$($a$)$ Give the general formula for the present value $P$ of an annuity growing at

the rate $t\%$ and with $r\%$ interest rate lasting $N$ years, having payments

which begin at $S$ Euros in the first year.

$($b$)$ Find the present value of a growing annuity, lasting $10$ years, having

payments which begin at $2,000$ in the first year and grow at a rate of

$3\%$ annually thereafter, and subject to an annual interest rate of $4\%.$

$($c$)$ If the annual interest rate $4\%$ in $($b$)$ is changed to $5\%,$ is the present value

lower or higher?

$($d$)$ Find the present value of a perpetual revenue stream having constant

flow rate

$R(t)=2,$

which is subject to a fixed interest rate of $3\%.$ :