Solving Geometric Sequences: Finding Time to Pay Off Mortgage

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Homework Help Overview

The problem involves calculating the time required to pay off a mortgage of £80,000 through annual payments of £5,000, with an interest rate of 4% applied to the outstanding balance. The context centers around geometric sequences and their application in financial calculations.

Discussion Character

  • Exploratory, Assumption checking, Mathematical reasoning

Approaches and Questions Raised

  • Participants discuss the iterative process of subtracting payments and adding interest, questioning how to express this in terms of a geometric sequence. Some participants are exploring the formula for the sum of a geometric series and its relevance to the problem.

Discussion Status

The discussion is ongoing, with participants sharing their thoughts on the calculations and questioning the accuracy of each other's results. There is no clear consensus on the correct time to pay off the mortgage, and multiple interpretations of the problem are being explored.

Contextual Notes

Some participants note that without interest, the mortgage would take 16 years to pay off, indicating that the interest will extend the repayment period. There is also mention of specific formulas that may apply to the situation, but no definitive conclusions have been reached.

seboastien
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Homework Statement


A mortgage is taken out for £80,000. It is to be paid by annual instalments of 5000with the first payment being made at the end of the first year that the mortgage was taken out. Interest of 4% is then charged on any outstanding debt. Find the total time taken to pay off the mortgage.


Homework Equations


Sn=a(r^n-1)/r-1


The Attempt at a Solution



I have no idea here can someone please point me in the right direction, thanks!
 
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seboastien said:

Homework Statement


A mortgage is taken out for £80,000. It is to be paid by annual instalments of 5000with the first payment being made at the end of the first year that the mortgage was taken out. Interest of 4% is then charged on any outstanding debt. Find the total time taken to pay off the mortgage.

Homework Equations


Sn=a(r^n-1)/r-1

The Attempt at a Solution



I have no idea here can someone please point me in the right direction, thanks!

Since you want a pointer in the right direction, start with your 80K subtract the payment 5K...then add to what remains 4% interest, then subtract 5K again and add to what remains 4% interest, then subtract...and so on. (so long as you don't actually calculate anything here you should see that you'll be summing up terms in a geometric sequence)
Clearly if you keep repeating this process the debt will become zero. the n in your relevant equation is the number of times interest gets added.
For what value of n will it be true that your above summation will be equal to zero?
 
Last edited:
I'm getting just over 12 years, is that correct?
 
it doesn't sound right
 
with no interest, paying 5000/year. it would take 80000/5000 = 16 years to pay it off. it's going to be more than that with interest
 
seboastien said:
I'm getting just over 12 years, is that correct?

what expression did you form to get that?
furthermore, shouldn't your sum equation be of the form: Sn = (r^(n+1)-1)/(r-1)?

Hmm..am I right in saying that you got 12 years by the following process:
1.a): 80000-5000
1.b): (1.a) - 0.04*(1.a)
1.c): (1.b) - 5000
1.d): (1.c) - 0.04*(1.c)...and so on?
 
Last edited:

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