Compound Interest Question. Any Help

Question

The 50 000 USD you have saved up by just paying 1000 USD into your account every 1-st of the month for a number of months. The account pays you interests at the rate of 3 % p.a compounded monthly.

Calculate now how many months are required to save up the 50 000 USD.

My solution:

Let us guess 49 months and we get:

1000*(1.03)*49 = 50 470

So after 49 months, I save up 50 470 USD

Anything wrong? Thanks

Answer 1

Each month your earn interest on the previous months balance.

Or you have one payment that receives interest every month. One payment that receives interest every month but the first. One payment that receives interest every month but the first two. etc.

$B = P\sum_\limits{k=1}^{n} (1+y)^n\\ 50,000 = 1000\sum_\limits{k=1}^{n} (1.0025)^n$

Now you need to sum a finite geometric series to find $n.$

Answer 2

$S=\$\,50,000$, $P=\$\, 1,000$ and $i=\frac{i^{(12)}}{12}=\frac{3\%}{12}=0.25\%$.

Using the future value of an annuity due (payment at the beginning) $$ S=P\,\ddot s_{\overline{n}|i}=P\cdot(1+i)\cdot\frac{(1+i)^n-1}{i} $$ we have $$ n=\frac{\log\left(\frac{S}{P}\frac{i}{1+i}+1\right)}{\log(1+i)}\approx 47 $$