You take out a $100,000 mortgage that involves monthly payments of $500 for 7 years (the first payment one month from now) and then a single lump sum payment in month 85. If the interest rate on the loan is 8% APR compounded monthly, how big is the single lump sum payment?

The lump sum in month 85 is $119,585.

Explanation:

We have: Discount rate = The monthly interest rate = APR / 12 = 8%/12 = 0.667%;

Denote x is the lump sum needs to be found.

We have the present value of 85 repayments discounted at monthly interest rate of 0.667% should be equal to $100,000 which the the present value of the mortgage.

As a result, we have the below calculations:

($500/0.667%) x (1 - 1.00667^-84) + x/1.00667^85 = $100,000

0.568x = $67,924 x = $119,585.

So, the lump sum in month 85 is $119,585.