Question about regular saving compound interest formula I have an exercise for my economics course in school where we have already the solution from our teacher received but we do not have the proof on how you actually solve this exercise.
I tried a lot of formulas and sites that I found, including this one:
http://www.calcudora.com/regular-savings-calculator.php
But I can not seem to find on how to make the formula to find the solution?
The exercise is as follows:

You deposit each month 25€ on a savings account, and you get a return of 7,5%. How much does the total amount include after 10, 20 and 30 year?

Now the solution of our teacher explains:
"If you deposit each month 25€ with a return of 7,5% on an annual basis which gets included at the end of each year and after it gets capitalized at the same 7,5%, then the total amount after 10 years will be 4416,54€.
But whatever formula I tried, I couldn't get to the 4416,54€.
I hope somebody knows how to solve this and wants to help me?
Ps: Sorry for my english if its bad, I am a belgian student and still working at my english language.
Screenshot of the solution from that site here mentioned above
 A: The last deposit is worth $25$, the one before that $25(1+\frac{0.75}{12})$, the one before that $25(1+\frac{0.75}{12})^2$ up to $25(1+\frac{0.75}{12})^{119}$.  This is a geometric series that we can sum, getting $$25\frac {(1+\frac{0.075}{12})^{120}-1}{\frac {0.075}{12}}\approx 4448.26$$
This assumes that deposits are made at the end of each month.  I had Alpha do the computation and also did it with my own spreadsheet which agreed to better than nine places.
A: You have a very special case here. You compound annually but the deposits are made monthly. What the calculator does is to compound the monthly payments every month but uses simple interest. The interest of the first payment  at the end of the year is $0.075\cdot\frac{12}{12}\cdot   25$. The interest of the second payment  at the end of the year is $0.075\cdot \frac{11}{12}\cdot 25$. The interest of the third payment  at the end of the year is $0.075\cdot \frac{10}{12}\cdot 25$ and so forth.
Summing up the fractions: $\sum_{i=1}^{12} \frac{i}{12}=\frac{1}{12}\cdot \sum_{i=1}^{12} i=\frac{1}{12}\cdot \frac{12\cdot 13}{2}=6.5$
Thus after one year there are $C_1= 300+25\cdot 0.075\cdot 6.5$
And after 10 years there are $C_{10}=(300+25\cdot 0.075\cdot 6.5)\cdot \frac{1-1.075^{10}}{1-1.075}=4416.54\ €$
