How to solve annuity problems?

Question

Rachael deposits $3,600 into a retirement fund each year. The fund earns 7.5% annual interest, compounded monthly. If she opened her account when she was 20 years old, how much will she have by the time she’s 55? How much of that amount was interest earned?

I solved through finding monthly compounding rate 1.00625. Then (3600-3600*((1.00625^12)^35))/(1-1.00625^12) Where is my mistake can someone help me with it?

Answer 1

If we start a year with $N$ dollars, we earn $r(N) = N\cdot(1+\frac{15}{2\cdot 100\cdot12})^{12} - N = (1.00625^{12}-1)N$ in interest over that year.

So at year $0$ (start), she puts in 3600. Compute $r(3600)$ and add it to the capital of $3600$. Then she puts in a new $3600$. So we have $7200 + r(3600)$ at the start of year $1$. So she earns $r(7200+r(3600))$ in interest.

At the start of year 2 we have $7200+r(3600) + r((7200+r(3600))$ to which we add another $3600$, etc. Keep on computing and add the interests to see what we earn in total.

Or in Python:

def rente(N,r):
      a=1+r/12
      return ((a**12)-1)*N

storting=3600
totaalrente=0
kapitaal = 0
r = 0.075

for i in range(35):
      kapitaal += storting
      e= rente(kapitaal,r)
      totaalrente += e
      kapitaal += e

print kapitaal, totaalrente