Hi, I'm pretty sure I did this right, I am in my 3rd year as an actuarial science student and studying for the Actuarial FM exam.

If your cash flows decrease every year by 4%, that means your cash flows are 96% of what they were each year. (100% - 4% = 96%)

So initially your cash flow is 1000, then after year one is 1000(.96), after year 2 is 1000(.96)(.96) and after year 3 is 1000(.96)(.96)(.96) so on and so forth. Rewritten the sum of the cash flows is:

1000 + 1000(.96) +1000(.96)^{2} +1000(.96)^{3} ...

The formula for a sum of geometric progression is a(1 - r^{m}) / (1-r)

so then sum of the initial and 23 following cash flows = 1000(1 - .96^{23}) / (1- .96)

the sum of the initial and 22 following cash flows = 1000(1 - .96^{22}) / (1 - .96)

If we subtract the 2 we should get the amount of the 23rd year cash flow.

Part 2. use the same formula but use 50 for *m*. 1000(1 - .96^{50}) / (1 - .96)

My answers: 15,223.61 - 14,816.26 = 407.35 cash flow after 23 years

21,752.86 sum of the cash flows after 50 years