Annuity variants

(a) Find the PV of a growing annuity whose $n$ payments $c, c(1+g), \dots$ arrive at $t = 0, 1, \dots, n-1$ (an annuity due). (b) Show that if each year's cash flow arrives at mid-year rather than year-end, every DCF term is multiplied by $(1+r)^{1/2}$. (c) Derive the continuous-time analogue $\int_0^T c\,e^{gt}e^{-rt}\,dt$ and its limit as $T \to \infty$.