$$ Annunity_{t} = C_{1} \cdot (1 + g)^{t-1} $$
$$ PV = \frac{C_{1}}{r - g} \cdot \left(1 - \left[\frac{{(1 + g)}}{{(1 + r)}}\right]^t\right) $$
Proof $$ PV = C_{1} \cdot \frac{(1+g)^0}{(1+r)^1} + C_{1} \cdot \frac{(1+g)^1}{(1+r)^2} + ... + C_{1} \cdot \frac{(1+g)^{t-1}}{(1+r)^t}\qquad (1) $$ Multiply both sides of (1) by (1+r)/(1+g): $$ PV \cdot (\frac{1+r}{1+g}) = C_{1} \cdot \frac{1}{(1+g)} + C_{1} \cdot \frac{1}{(1+r)} + C_{1} \cdot \frac{1+g}{(1+r)^2} ... + C_{1} \cdot \frac{(1+g)^{t-2}}{(1+r)^{t-1}}\qquad (2) $$ (1) - (2): $$ PV \cdot (1 - \frac{1+r}{1+g}) = C_{1} \cdot (\frac{(1+g)^{t-1}}{(1+r)^{t}}) - C_{1} \cdot (\frac{1}{1+g}) $$ $$ PV = C_{1} \cdot (\frac{1+g}{g-r})\left[ \frac{(1+g)^{t-1}}{(1+r)^t} - \frac{1}{(1+g)} \right] $$ $$ PV = \frac{C_{1}}{r - g} \cdot \left(1 - \left[\frac{{(1 + g)}}{{(1 + r)}}\right]^t\right)\qquad (3) $$
$$ FV = C_{1} \cdot \frac{{(1 + r)^t - (1 + g)^t}}{{r - g}} $$
Proof
Compounding PV in (3) to FV:
$$
FV = PV (1+r)^t
$$
$$
FV = \frac{C_{1}}{r - g} \cdot \left(1 - \left[\frac{{(1 + g)}}{{(1 + r)}}\right]^t\right) \cdot (1+r)^t
$$
$$
FV = C_{1} \cdot \frac{{(1 + r)^t - (1 + g)^t}}{{r - g}}
$$