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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

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    Including words like "desperate" in your title are less likely to gather answers. – 2017-01-19
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    I just made a spreadsheet and assumed the deposits are at the end of the month. It gives $4448.26$ at the end of $120$ months, close but not exact. Even rounding the interest down to the penny doesn't get me there. – 2017-01-19
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    @RossMillikan I also got that value once, and other values close by, but apparently the solution is really correct and it should be possible to find it. That is all I know from it. :( – 2017-01-19

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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 $C_{10}=(300+25\cdot 0.075\cdot 6.5)\cdot \frac{1-1.075^{10}}{1-1.075}=4416.54\ €$

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    @Filip: If that is what is desired, the problem setter owes you that explanation. It would be much more common to compound the interest every month. Simple interest over the term would also be believable, but the hybrid is very strange. – 2017-01-20
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    @callculus: If I try your solution and put in my calculator or wolframalpha I get 6313 as solution. Or is there something I did wrong for input? http://www.wolframalpha.com/input/?i=(1%2B0.075%E2%8B%856.5)%E2%8B%85300%E2%8B%85((1%E2%88%921.075%5E10)%2F(1%E2%88%921.075)) – 2017-01-20
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    @Filip You are right. I have calculated the interest for the 25€ deposit by using 300€, which was clearly wrong. I fixed it. – 2017-01-20
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    @Filip To make it easy for you to proof it here is the link: http://www.wolframalpha.com/input/?i=(300%2B25%5Ccdot+0.075%5Ccdot+6.5)%5Ccdot+%5Cfrac%7B1-1.075%5E%7B10%7D%7D%7B1-1.075%7D – 2017-01-20
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    @callculus I see it now. Many thanks, you are a life saver. – 2017-01-20
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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.