# Math Help - Investment Problem

1. ## Investment Problem

Hey can someone plz help on this problem.

Jason is 20 years old and his salary next year will be \$20000. His salary is going to increase at a rate of 5% each year till he is 50. How much money would he have saved by that time if he saves 5% each year and invest the savings at an interest rate of 8%?

Thanks

JT

2. There are some possible interpretive variations in your fact-set that might need clarifying, such as number and timing of the investment-deposits. But you can easily adjust the following basic approach as need be.

I'll set it up assuming Jason will make one deposit per year, consisting of 5% of his salary of that given year, and it will be made at the end of such year. Further, he'll have a total of 30 such deposits, and we'll compute his accumulated amount immediately following the final one.

The set-up lays out as....

$T\ =\ 20,000(0.05)(1.08)^{29}\ +\ 20,000(1.05)(0.05)(1.08)^{28}\ +\ ...\ +\ 20,000(1.05)^{29}(0.05)$

...where T is the total accumulated amount. Note that the FV of the first, second, and final deposit are shown explicitly, with the others implied.

Note that this is a Geometric Series, with initial amount a = $20,000(0.05)(1.08)^{29}$; common ratio r = $(1.05)(1.08)^{-1}$; and there are a total of n = 30 terms.

With that, use the sum-of-a-GS formula...

$T\ =\ \frac{a(1-r^n)}{1-r}$

...to quickly obtain Jason's final accumulated total.

4. Here's a way to "general case" your problem:

A = accumulated total (?)
S = initial salary (20000)
i = interest rate paid (.08)
j = rate (percentage) of salary deposited (.05)
k = rate of salary increase (.05)
n = number of years (30)

A = Sk{[(1 + i)^n - (1 + j)^n] / [i - j]

A = 20000(.05)[(1.08^30 - 1.05^30) / (.08 - .05)] = 191,357.15046....
(same results as with LochWulf's way)