Problem $8.$ A growing annuity is a stream of cash flows that pays at regular intervals and

growing at a constant rate $g>0,$ up to some final date. In particular, it pays $C(1+g{)}^{n-1}$

at the end of $n^{th}$ year until, where $C>0$ is a constant. Cash flows are paid until the end of

the $N^{th}$ year.

$($a$)$ Show that the present value of a growing annuity is

$PV=\frac{C}{r-g}(1-(\frac{1+g}{1+r}{)}^N)$

where $r$ is a discount rate.

$($b$)$ Does the above formula hold if $rg?$

Prove the formula in question $($a$)$ and answer question $($b$)$