How does one prove that:

$$\ddot{a}_\overline{n|} = 1 + v + v^2 + \cdots + v^n = \frac{1-v^n}{d}$$

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Well Simon, it seems you have included an extra R1 cashflow at the end of year \(n\). The annuity listed says that the last cashflow is received at year \(n-1\). Nevertheless, here is a proof!

Equation 1: \( PV = 1 + v + v^2 + v^3 +....+ v^{n-1}\)

Equation 2: \(v.PV = v + v^2 + v^3 + ...+ v^n\)

Equation 1 - Equation 2:

\( PV - v \cdot PV = 1 + v + v^2 + ...+ v^{n-1} - (v + v^2 + v^3 + ... + v^{n-1} + v^n) \)

\(= PV (1-v) = 1- v^n\)

\(\implies PV = \frac{1 - v^n}{1 - v}\)

And \((1 - v) = d\)

Therefore, \(PV = \frac{1-v^n}{d} \)