Question

A 20-year loan is to be repaid with level payments at the end of each year. The amount of interest paid in the 5th payment is 103.17. The amount of interest paid in the 13th payment is 57. Find the amount of interest and principal paid in the 17th payment. (Answers: 30, 270.)

Answer 1

interest for 5th payment = yearly payment (pm) - principal in 5th payment

103.17 = pm(1 - v^(20 - 5 +1))

103.17 = pm( 1 - v^16)

interest for 13th payment = yearly payment (pm) - principal in 13th payment

57 = pm(1 - v^(20-13+1))

57 = pm(1 - v^8)

1st equation can be written as below

103.17 = pm * (1 - v^8) * (1 + v^8)

using a^2 - b^2 = (a - b) * (a + b)

now put the value of equation 2

103.17 = 57 * (1 + v^8)

v^8 = 103.17/57 - 1

v^8 = .81

square root of it will give

v^4 = .9 or -.9

interest for 17th payment = yearly payment (pm) - principal in 17th payment

interest for 17th payment = pm(1 - v^(20-17+1))

interest for 17th payment = pm(1 - v^4)*(1 + v^4)/(1 + v^4)

= 57/(1.9)

= $30

Interest for the 17th payment will be $30.