Respuesta :
We have an initial balance of $5500.
The APR is 24% and each month only 2% of the balance (the minimum payment) is paid.
An APR of 24% means an actual monthly interest rate of:
[tex]i=\frac{\text{APR}}{12}=\frac{0.24}{12}=0.02[/tex]Then, the remaining balance each month will be increased by 0.02 or 2% because of the interest.
We also have to substract from the balance the 2% payment done every month.
Calculation of the first balance:
We start with a balance of $5500. It will generate an interest of 2% each month that is equivalent to 0.02*5500 = 110.
Then, the balance for the first month will be 5500+110 = $5610.
If we pay 2% of this amount, we pay 0.02*5610 = 112.2.
This will left a balance of 5610-112.2 = $5497.80.
Generalization:
We can calculate the balance for the first month, B(1), as:
[tex]\begin{gathered} B(1)=(B(0)+I)\cdot(1-p) \\ B(1)=B(0)\cdot(1+i)\cdot(1-p) \end{gathered}[/tex]We can replace i, the monthly interest rate, and p, the payment percentage, and rearrange this as:
[tex]\begin{gathered} B(n)=B(n-1)\cdot(1+i)(1-p) \\ B(n)=B(n-1)\cdot(1.02)(0.98) \\ B(n)=B(n-1)\cdot0.9996 \end{gathered}[/tex]Then, we see that each month the balance will be reduced to a 99.96% of the balance of the previous month.
Then, we can extrapolate this to any month as:
[tex]\begin{gathered} B(1)=B(0)\cdot0.9996 \\ B(2)=B(1)\cdot0.9996=B(0)\cdot0.9996\cdot0.9996=B(0)\cdot0.9996^2 \\ B(n)=B(0)\cdot0.9996^n \\ B(n)=5500\cdot0.9996^n \end{gathered}[/tex]If we calculate the balance for the 30th month, we get:
[tex]\begin{gathered} n=30\Rightarrow \\ B(30)=5500\cdot0.9996^{30} \\ B(30)\approx5500\cdot0.9881 \\ B(30)\approx5434.38 \end{gathered}[/tex]Answer: the balance after 30 months will be approximately $5434.38
