A monthly deposit savings online calculator with a monthly compounding is presented.

Let

\( a_1 \quad , \quad R \times a_1 \quad , \quad R^2 \times a_1 \quad .... \quad R^n \times a_1 \)

be the term of a geometric sequence.

The geometric sequence sum \( S_n \) defined by

\( S_n = a_1 + R \times a_1 + R^2 \times a_1 .... R^n \times a_1 \)

is given by the formula

\[ S = a_1 \dfrac{1 - R^{n+1}}{1 - R} \qquad (I) \]
\( a_1 \) is the first term and \( R \) is the common factor.

This calculator helps you find how much you need to save each month (monthly deposit) in order to reach a saving goal \( TS \) in \( n \) years at an annual interest rate \( r \) with monthly compouding.

We first write r as a decimal number

\[ r_d = \dfrac{r}{100} \]
Since \( r \) is an annual rate, the monthly rate is
\[r_m = \dfrac{r_d}{12}\]

Let \( MD \) be the monthly deposit and r be the annual rate on interets.

- Initial (first) deposit is MD.

- At the end of the first month \( (n = 1) \), the total savings is given by

\( TS = MD + r_m MD = MD \times (1 + r_m) \)

Then \( MD \) is deposited, hence \( TS = MD + MD \times (1 + r_m) \)

- After the end of the second month \( (n = 2) \), the total savings is given by

\( TS = MD \times (1 + r_m) + MD \times (1 + r_m)^2 \)

Then \( MD \) is deposited, hence \( TS = MD + MD \times (1 + r_m) + MD \times (1 + r_m)^2 \)

- At the end of the third month \( (n = 3) \), the total savings is given by

\( TS = MD \times (1 + r_m) + MD \times (1 + r_m)^2 + MD \times (1 + r_m)^3 \)

and so on

- After the end of the nth month \( (n = n) \), the total savings is given by

\( TS = MD \times (1 + r_m) + MD \times (1 + r_m)^2 + MD \times (1 + r_m)^3 + ... + MD \times (1 + r_m)^n\)

Factor \( MD \) out and rewrite \( TS \) as a geometric series sum.

\( TS = MD \; \left((1 + r_m) + (1 + r_m)^2 + (1 + r_m)^3 + ... + (1 + r_m)^n \right) \)

The sum inside the parentheses is a geometric sequence sum whose first term is \( 1 + r_m\) and its common factor is \( 1 + r_m \), hence using the formula for a geometric sequence sum , \( TS \) is given

\( TS = MD \times (1 + r_m) \dfrac{1 - (1 + r_m)^{n}}{ 1 - (1 + r_m) } \)

which simplifies to

\[ TS = MD \times \dfrac{(1 + r_m)^{n+1} - 1 - r_m}{ r_m } \]

If we know our target to reach after \( n \) months and we need to calculate the monthly deposit MD, we have \[ MD = TS \times \dfrac{r_m }{(1 + r_m)^n - 1-r_m} \] The above formula is used in the calculator below.

Enter the savings goal which the total amount of money you need to save which in the formula above is \( TS \), the annual rate of intrest \( r \) and the number of years.

The calculator gives you the monthly deposit needed to achieve your goal which savings a certain amount.

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Years