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Question: How much money will you have at the end of four years if you deposit $1,000 a year (at the end of each year) into a bank account paying 10% interest?

We could answer this by calculating the future value of each individual deposit:

$$1000 *( 1.1^0 + 1.1^1 + 1.1^2 + 1.1^3 ) = 4641 = 1000 * \sum_1^T (1+r)^{T-1}$$

We will have a total of $4,641 in the bank at the end of four years.

Then we also know:

Future Value = Annuity Cash Flow * Future Value Annuity Factor

Future Value Annuity Factor = $\frac 1r[(1+r)^n -1]$

So Why $\frac 1r[(1+r)^n -1] = \sum_1^T (1+r)^{T-1}$ ??

How can we deduce from left side term to the right side term??

Thanks a lot.

Micah
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Emily
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1 Answers1

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If I recall you can use geometric series.

$\displaystyle\sum_{k=0}^{n-1}C(1+r)^k$. Note that this is a geometric series. If the cash flow is constant then this is $\\ C\displaystyle\sum_{k=0}^{n-1}(1+r)^k=C\left[\frac{1-(1+r)^n}{1-(1+r)}\right]=C\left[\frac{1-(1+r)^n}{-r}\right]=C\left[\frac{(1+r)^n-1}{r}\right]$

GAM
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