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 Question from Robin, a student: A person is going to arrange a loan at 3 mill. he will pay 150000 back each year , so it take 20 yrs. to finish. In addition he pays 5% interest of what is leaft each year. How do I find the sum-function which shows the total sum he have payed?

Hi Robin.

Let's write out the first few years to examine the pattern closely:

 End of year number Balance before payment Fixed Payment Balance carried forward 5% Interest on Balance carried Cumulative Payment Total 1 3 000 000 150 000 2 850 000 142 500 192 500 2 2 850 000 150 000 2 700 000 135 000 577 500 3 2 700 000 150 000 2 550 000 127 500 855 000

The cumulative payment total (CPT) is the previous CPT + the fixed payment + the interest for the current year.

The interest for year n is the balance carried forward times 5%. The balance carried forward for year n is $3 000 000 minus$150 000 times n.

So CPT(n) = CPT(n-1) + 150 000 + 0.05 * (3 000 000 - 150 000n).

Another way of writing this is by using summation. Remember that a sum of a sum is a sum, so we can simplify the summation in steps:

Now the remaining summation is just the sum of the first few natural numbers. This is a well-known sum: the sum of the first A natural numbers is equal to A(A+1)/2, so we complete the problem as follows:

There's the result. A total of \$4 425 000 would be paid out.

Math Central is supported by the University of Regina and The Pacific Institute for the Mathematical Sciences.