proof of present value of annuity using the formula for the sum of geometric series

Question

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Hi, so I have the steps for deducing the present value of annuity from the formula for the sum of the geometric series. However, I don't seem to understand step a,b and c. I know how to deduce the present value of annuity, but not in this way.

Answer 1

It's just horrid notation. They use some symbols twice with different meanings. If you add new symbols, the confusion disappears: what happens is that they have a formula for a geometric sum which is

$$S_n=\frac{a(1-x^n)}{1-x} \; .$$

And they now that $A$ is expressed as a geometric sum, so identifying the ratio $x$ and the first term $a$:

$$a = R(1+r)^{-1}$$

and

$$x=(1+r)^{-1} \; ,$$

they can conclude that

$$A = \frac{R(1+r)^{-1}(1-(1+r)^{-n})}{1-(1+r)^{-1}} \; .$$

The rest is just algebraic manipulations until the end result.