Compound interest with monthly fees

Question

> A bank account pays interest at 12% compounded monthly, and has a
> monthly fee of $10, deducted at the end of each month. If $13,000 is
> deposited on January 1, 2013, how much is in the account on January 1,
> 2016?

I have solved similar questions but none ever had fees.

Thanks

Answer 1

The fee is just a negative deposit. It draws (negative) interest just like a positive deposit. I assume the $12\%$ is annual interest, so the monthly interest is $1\%$. Then at the end of month one you have $13000\cdot 1.01 - 10$ At the end of month two you have $13000 \cdot 1.01^2 - 10 \cdot 1.01 -10$ Each month, you raise the exponents by $1$ and add one more $-10$ at the end. All the fees become a geometric series which you can sum, getting a closed form. Do you know how?

Added: At the end, you will have $13000\cdot 1.01^{36}-10\sum_{n=0}^{35}1.01^n$. Now you need to do the last sum.