Monthly Savings Calculator

Review Geometric Sequence Sum

Let \[ a_1, \quad R \times a_1, \quad R^2 \times a_1, \quad \dots, \quad R^n \times a_1 \] be the terms of a geometric sequence.

The geometric sequence sum \( S_n \) defined by \[ S_n = a_1 + R \times a_1 + R^2 \times a_1 + \dots + R^n \times a_1 \] is given by the formula:

where \( a_1 \) is the first term and \( R \) is the common factor.

Monthly Deposit Savings Formula

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

We first write \( r \) as a decimal number: \[ r_d = \frac{r}{100} \] Since \( r \) is an annual rate, the monthly rate is: \[ r_m = \frac{r_d}{12} \]

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

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 another \( MD \) 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 another \( MD \) 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: \[ TS = MD \times (1 + r_m) + MD \times (1 + r_m)^2 + MD \times (1 + r_m)^3 \quad \text{and so on.} \]
5. After the end of the \( n \)-th month, the total savings is given by: \[ TS = MD \times (1 + r_m) + MD \times (1 + r_m)^2 + MD \times (1 + r_m)^3 + \dots + MD \times (1 + r_m)^n \] Factoring out \( MD \), we rewrite \( TS \) as a geometric series sum: \[ TS = MD \left( (1 + r_m) + (1 + r_m)^2 + (1 + r_m)^3 + \dots + (1 + r_m)^n \right) \] The sum inside the parentheses is a geometric sequence sum whose first term is \( 1 + r_m \) and common factor is \( 1 + r_m \). Using formula (I): \[ TS = MD \times (1 + r_m) \frac{1 - (1 + r_m)^n}{1 - (1 + r_m)} \] which simplifies to: \[ TS = MD \times \frac{(1 + r_m)^{n+1} - 1 - r_m}{r_m} \] Solving for the required monthly deposit \( MD \): \[ MD = TS \times \frac{r_m}{(1 + r_m)^n - 1 - r_m} \]

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