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?

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.

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