To determine the correct formula for calculating the monthly mortgage payment for a fixed-rate mortgage, let's break down the standard formula and compare it with the given options.
The standard formula for calculating the monthly mortgage payment (M) is:
[tex]\[ M = P \cdot \frac{R (1 + R)^n}{(1 + R)^n - 1} \][/tex]
Where:
- [tex]\( M \)[/tex]= Monthly payment
- [tex]\( P \)[/tex] = Principal amount (the initial loan amount)
- [tex]\( R \)[/tex] = Monthly interest rate
- [tex]\( n \)[/tex] = Total number of monthly payments
Now, let's analyze each provided formula:
1. [tex]\( M = P \cdot \frac{R (1 - R)^n}{(1 + R)^n} \)[/tex]
This formula is incorrect. The term [tex]\((1 - R)^n\)[/tex] should not be used. The correct formula involves [tex]\((1 + R)^n\)[/tex] in both the numerator and the denominator.
2. [tex]\( M = P \cdot \frac{R (1 + R)^n}{(1 + R)^n - 1} \)[/tex]
This formula matches the standard formula for calculating the monthly mortgage payment. It correctly includes [tex]\((1 + R)^n\)[/tex] in both the numerator and the denominator, and the minus one in the denominator is correctly placed.
3. [tex]\( M = P \cdot \frac{R}{(1 + R)^n - 1} \)[/tex]
This formula is incorrect. It does not include the term [tex]\((1 + R)^n\)[/tex] in the numerator, which is necessary for the correct calculation.
4. [tex]\( M = P \cdot \frac{R (1 + R)^n}{n + R} \)[/tex]
This formula is incorrect. The denominator should be [tex]\((1 + R)^n - 1\)[/tex], not [tex]\(n + R\)[/tex].
Given these comparisons, the correct formula to use is:
[tex]\[ M = P \cdot \frac{R (1 + R)^n}{(1 + R)^n - 1} \][/tex]
From the provided formulas, this corresponds to:
[tex]\[ M = P \cdot \frac{R (1 + R)^n}{[(1 + R)^n - 1]} \][/tex]
So, the correct option is:
[tex]\[ \boxed{M=P \frac{\left[R(1+R)^n\right]}{\left[(1+R)^n-1\right]}} \][/tex]
