# Construct an equation that models the repayment of a high value loan. Such as a mortgage. The loan is based on compound interest with a monthly calculation and repayment cycle. The site system forcing me to add a question mark: ?

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Set the monthly repayment as R, the annual percent interest (apr) as T% and the initial principle sum as #P_0#

This question is set so that I may demonstrate some mathematical processes.

Set the monthly repayment as R, the annual percent interest (apr) as T% and the initial principle sum as

This question is set so that I may demonstrate some mathematical processes.

##### 2 Answers

Let the number of years be

Let the count of calculation cycles be

Given that the initial principle sum is

Set the adjusted principle after the 1st cycle as

Set the adjusted principle after the 2nd cycle as

Set the adjusted principle after the 3rd cycle as

The interest for 1 year is

So splitting this over each month gives

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Repeating this process but with full substitution

..........................................................................

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Using the same approach we end up with:

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Let

Factor out the

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From this it is obvious that ( hate that phrase!)

For any n we have:

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With full explanation this is a big solution so I am splitting it

See Solution part 1 first

Following on from:

Set as

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Note that at the completion of paying off the loan

To take this further we need to determine the sum of the series within the brackets.

set

Then

Factor out the

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Substitute

But

To determine the different values set

Do not forget that