A man repays a loan of 65,000 by paying 400 in the first month and then increasing the payment by 300 every month. How long will it take for him to clear the loan?

Solution :

For the first month he is paying = 400

payment of second month = 400 + 300 = 700

payment of 3rd month = 700 + 300 = 1000

400 + 700 + 1000, ................

loan amount = 65000

(n/2)[2a + (n - 1)d] = 65000

(n/2)[2(400) + (n - 1)300] = 65000

(n/2)[800 + 300n - 300] = 65000

(n/2)[500 + 300n] = 65000

(n/2)[5 + 3n] = 650

n[5 + 3n] = 1300

5n + 3n2 = 1300

3n2 + 5n - 1300 = 0

3n2 - 60n + 65n - 1300 = 0

3n(n - 20) + 65(n - 20) = 0

(n - 20) (3n + 65) = 0

n = 20 or n = -65/3 (not acceptable)

Hence he will clear the loan amount in 20 months.