0.8 Mathematics of finance  (Page 6/9)

Again, suppose that $x$ dollars are deposited each quarter in the sinking fund. After five years, the future value of the fund should be $450,000. Which suggests the following relationship:

There are 60 deposits made in this account. The first payment stays in the account for 60 months, the second payment for 59 months, the third for 58 months, and so on.

The first payment of $500 will accumulate to an amount of $\$\text{500}{\left(1+\text{.}\text{08}/\text{12}\right)}^{\text{60}}$ .

The second payment of $500 will accumulate to an amount of $\$\text{500}{\left(1+\text{.}\text{08}/\text{12}\right)}^{\text{59}}$ .

The third payment will accumulate to $\$\text{500}{\left(1+\text{.}\text{08}/\text{12}\right)}^{\text{58}}$ .

And so on.

The last payment is in the account for a month and accumulates to $\$\text{500}\left(1+\text{.}\text{08}/\text{12}\right)$

To find the total amount in five years, we need to find the sum of the following series.

Written backwards, we have

If we add $500 to this series, and later subtract that $500, the value will not change. We get

Not considering the last term, we have a geometric series with $a=\$\text{500}$ , $r=\left(1+\text{.}\text{08}/\text{12}\right)$ , and $n=\text{60}$ . Therefore the sum is