Question

$1000 is invested at the beginning of each year for 10 years. The rate of interest is fixed at 7.5% per annum. Interest is compounded annually. Calculate the total value of the investments at the end of the ten years.

Answer 1

First off, find the rate of interest for 1 year.

Answer 2

Well, the amount you have after a year.

Answer 3

Are you following?

Answer 4

Yes, It'd be (1000)(1.075)=1075

Answer 5

Right.

Answer 6

Now you find 7.5% of 1075 so it gives you the amount for the second year and you do it for 10 years.

Answer 7

But that is not what the question is asking. After the first year I add another 1000 and then add 7.5% of the new total. So far what I've got is the following:

Answer 8

AHh you're right

Answer 9

So you have 1075 + 1000

Answer 10

\[ \left[ \left[ (1000*1.075)+1000 \right]1.075 + 1000 \right]1.075+1000.....\]
\[1000(1.075)^{10} + 1000(1.075)^{9} + 1000(1.075)^{8}......\]

Answer 11

So it should be a 10th sum of a geometric series, but when I use the formula I get:
\[\frac{ 1000(1-(1.075)^{10}) }{ 1-1.075 }\] = 14147
But the answer sheet says it must be 15208

Answer 12

Let me see.

Answer 13

Is it not a Power Series?
\[\sum_{i=0}^{\infty}a_{i}z^i\]
But instead of going to infinity go up to 10 and i starts at 1. Make a=1000. z=1.075.
\[\sum_{i=1}^{10}1000(1.075)^i=1000(1.075)^1+1000(1.075)^2+...+1000(1.075)^{10}\]Which ends up being 15208.

Answer 14

Thanks alot! I just assumed it would be a geometric series...

Answer 15

No problem. Took a while to figure that one out haha.