
Decreasing annuity
A loan is repaid by means of a decreasing annuity payable annually in arrears for 15 years.
The instalment at the end of the first year is £4,000 and subsequent instalments are reduced by
£150 each year. The rate of interest used is 12% p.a. effective.
a) Calculate the original amount of the loan.
b) Construct the capital/interest schedule for year 6, showing the outstanding capital at the
beginning and end of the year and the interest and capital components of the instalment.
c) Immediately after the sixth instalment, the interest rate on the outstanding loan is reduced
to 8% p.a. effective.
Calculate the amount of the seventh instalment if subsequent instalments are still to be
reduced by £150 each year and the loan is to be repaid by the original date (i.e. 15 years from the commencement).

Well, do it!
You have i = 0 .12, then v = 1/1.12
This gives: 4000v + (4000  150)v^2 + (4000  150*2)v^3 + ... + (4000  150*14)v^15 as the original value of the loan.
A little algebra gives:
4000v + 4000v^2 + 4000v^3 + ... + 4000v^15  (150*v^2 + 150*2*v^3 + ... + 150*14*v^15)
Just a hair more algebra helps quite a bit.
4000v(1 + v + v^2 + ... + v^14)  150*v^2(1 + 2*v + ... + 14*v^13)
What say you? Can you add it up from there?

how about the other parts of the question