in common terms the present value of a regular

Question

In common terms the present value of a regular annuity may be shown as given below:

PVNn = A/(1 + k) + A/(1 + k)²+ ..................+ A/(1 + k)^N

= A (1/(1 + k) + 1/(1 + k)²+ ..................+ 1/(1 + k)^N)

= A [((1 + k)^N- 1)/(k (1 + k)ⁿ)]

In case of annuity due:

PVA_{n (due )} = A [(1 + k)n  - 1)/(k (1 + k)ⁿ)] (1 + k)   ...........................Eq(12)

Here PVA_n = Present value of an annuity that has a duration of n periods;

A = Constant periodic flows and k = discount rate