Sum of Geometric progression?

Question

A man deposits $200 at the beginning of every year into a bank account at a compund interest rate of 3%per annum. Find out how much he has at the end of 10th year to the nearest dollar?

I honestly don't quite understand the part about "depositing $200 every year".

How do you go about solving this?

Answer 1

The first $\$200$ compounds for $10$ years the next year he deposits another $\$200$ into the account, and that will compound for $9$ more years.

After 10 years he has:

$200(1.03)^{10} + 200(1.03)^9 + \cdots + 200(1.03)$

And how would you sum that up?

Answer 2

At the end of the first year it's going to be $200 \cdot 1.03$;
at the end of the second year it's going to be $(200 \cdot 1.03 + 200) \cdot 1.03=200 \cdot (1.03+1.03^2)$;
at the end of the third year we will have $200 \cdot (1.03+1.03^2+1.03^3)$.
Do you see the pattern?