Geometric Progression: Finding the Outstanding Loan Amount after Each Year

[JJYEO325]

Question:
John took a bank loan of $200000 to buy a flat. The bank charges an anual interest rate of 3% on the outstanding loan at the end of each year. John pays $1000 at the beginning of each month until he finishes paying for his loan.
Let Un denote the amount owed by john at the end of the nth year,

(i)show that Un= k(u n-1 - 1200) where k is a constant to be determined.
(ii) Express u n in the form of a +(1.03^n)b, where a and b are constants to be determined.

Homework Equations

The Attempt at a Solution

year total amt owed (before end of year) ''(at the end of year)
1 $200000 200000(1.03)- 12000
2 200000(1.03)-1200 (1.03)^2(200000)-12000(1.03)


1

 

[HallsofIvy]

Question:
John took a bank loan of $200000 to buy a flat. The bank charges an anual interest rate of 3% on the outstanding loan at the end of each year. John pays $1000 at the beginning of each month until he finishes paying for his loan.
Let Un denote the amount owed by john at the end of the nth year,

(i)show that Un= k(u n-1 - 1200) where k is a constant to be determined.
(ii) Express u n in the form of a +(1.03^n)b, where a and b are constants to be determined.

Homework Equations

The Attempt at a Solution

year total amt owed (before end of year) ''(at the end of year)
1 $200000 200000(1.03)- 12000

that last value is incorrect. "The bank charges an anual interest rate of 3% on the outstanding loan at the end of each year." At the end of the first year he will have paid 12000 so the outstanding loan is 200000- 12000= 188000. You should have 1.03(200000- 12000)= 1.03(200000)- 1.03(12000)

2 200000(1.03)-1200 (1.03)^2(200000)-12000(1.03)

At the end of the second year, the outstanding balance will be 1.03(200000)- 1.03(12000)- 12000 so the interest will be 1.03(1.03(200000)- 1.03(12000)- 12000)= 1.03^2(200000)- (1.03^2(12000)+ 1.03(12000)= 1.03^2(200000)- 12000(1.03+ 1.03^2).

At the end of the third year, the outstanding balance will be 1.03^2(200000)- 12000(1.03+ 1.03^2)- 12000= 1.03^2(200000)- 12000(1+ 1.03+ 1.03^2) so the interest will be 1.03(1.03^2(200000)- 12000(1+ 1.03+ 1.03^2)= 1.03^3(200000)- 12000(1.03+ 1.03^2+ 1.03^3).

Get the idea?

 

[JJYEO325]

Thanks.. So the total amount he owe the bank for the first year is Un >? which is 1.03(200000)-(1.03)12000? Sorry I don't seem to understand the question.