Hello I have just worked a question in which I get an answer different to the answer in my book.
The question states:
If a person deposits 500 at the end of each month for 20 years at an AER of 5% find the total sum after 20 years.
I first found the monthly interest to be $0.4074$%
Then used the following sum of a geometric formula:
$$500(\frac{1-(1.004074^{239})}{1-1.004074})$$
That went fine and I got an answer, I used 239 as the number of months as no interest will be received for the first month as the money is deposited at the END of each month.
The answer in my book can only be got by using 240 as the number of months, could someone explain the reasoning behind this.
Thanks
Let the monthly interest rate be $r$. The last payment will accrue no interest, so has "grown" to $500$. The one before that will have grown to $500(1+r)$. The next to last payment will have grown to $500(1+r)^2$. And so on. The first payment will have been in the bank for $239$ months, so has grown to $500(1+r)^{239}$. Thus the total is $$500(1+(1+r)+(1+r)^2+\cdots+(1+r)^{239}.$$ This geometric series has sum $$500\cdot \frac{(1+r)^{240}-1}{(1+r)-1}.$$