# loan repayment- finding the loan if end payment increases by certain amount

What happens if a loan has an effective annual interest rate $$i = 10$$% is repaid with 10 yearly payments starting one year after the loan. The amount of the first payment is $$500$$ but each subsequent payment is $$10$$ larger than the previous payment.

Similar to the other question I asked in loan repayment- find the loan and interest paid.

I found the loan using $$\require{enclose} L = 500\left(1-(1/1.1)^{10}\right)/0.1 =3072.283$$

I did a table to check what the outstanding balance will be at 10th payment but I get negative outstanding balance. Does that make sense?

• You can use Goal Seek under Tools to find the starting payment that yields a balance of $0$ at the desired time. Using the formula gives a very good starting value, which increases the chance you will find a solution. Mar 28 at 2:56

The remaining debt after each payment is calculated by first applying the interest then deducting the amount paid. Therefore the remaining debt after each payment is as follows:

1. $$P(1+i)-500$$
2. $$P(1+i)^2-500(1+i)-510$$
3. $$P(1+i)^3-500(1+i)^2-510(1+i)-520$$

and so on, up to the zero residual debt after last payment:

$$P(1+i)^{10}-500(1+i)^9-510(1+i)^8-...-580(1+i)-590=0$$

This residual zero debt helps calculate the loan P:

$$P=\frac{\sum_{k=0}^{9}{(500+10k)(1+i)^{9-k}}}{(1+i)^{10}}$$

Using i=10%, I’ve got P=3301.19

• I don't understand your formula for P. I have to use actuarial notation to calculate the loan. Mar 28 at 2:46
• The formula you’re using to calculate the loan is not correct because it accounts for equal payments. Your payments are increasing and my formula accounts for the increase as you see I get almost zero debt after the last payment Mar 28 at 2:55
• I have solved for P in the zero residual debt after last payment Mar 28 at 3:02