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.
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 $\\ C\displaystyle\sum_{k=0}^{n-1}(1+r)^k=C\left[\frac{1-(1+r)^n}{1-(1+r)}\right]=C\left[\frac{1-(1+r)^n}{-r}\right]=C\left[\frac{(1+r)^n-1}{r}\right]$