# Monthly Savings Calculator

       

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

## Review Geometric Sequence Sum

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.

## Monthly Deposite Savings Formula

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.

1. Initial (first) deposit is MD.
2. 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 $$TS = MD + MD \times (1 + r_m)$$

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

4. 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

5. 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.

## Use of Savings Calculator

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.

Dollars

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Years