Please show the working of present value through first principles as I was not able to get the question. I did it as [(100+100v+.......100v^19)+(120v^20+......+120v^24)+200v^25+.......+200v^39)]. Please correct me where I am wrong.
It looks like you've not allowed for the fact that you have monthly payments at all. Writing out the present value of montly payments from first principles seems a rather long-winded approach, and I would advise against it wherever possible. If you're trying to simplify what you have, pull out a factor of 120v^20 from your second brackets to give: 120v^20(1+v+...+v^4) The terms in brackets here are an annuity in advance with a term of 5. You could do something similar with your last set of payments (of 200).
