If a student makes monthly deposits of 1,200 into an account with a nominal annual interest rate of 4.5% compounded monthly, will he have enough after 5 years to purchase a $105,000 property in cash?
I already have the solution.
I just want to understand why did he use the following to find the annual effective interest rate
i = 4.5% / 12 = 3.75%
and the Number of compounding periods he used is 60
Why he didn't use the following standard formula:
If the nominal annual interest rate is 4.5%, then 4.5%/12 represents the monthly rate. The annual effective rate (compounding monthly) as per your formula is
$$(1+0.045/12)^{12}-1.$$
Now, as per the problem you mention, the timings are unclear, but let me give you a general formula (you can adjust accordingly to deal with different timings). If you make deposits of $x$ at the beginning of every period (starting immediately) and the per-period interest rate is $r$, observe that at the end of $t$ periods, your balance will be
$$x(1+r)^t+x(1+r)^{t-1}+...+x(1+r)=x(1+r)\frac{(1+r)^t-1}{r}.$$
Your setup is the case $x=1200,t=5*12=60,r=0.045/12.$