Original Amount

I don't understand your solution.

Look at my example again. Do you see how the answer after one year goes into the parentheses for year two?

I don't.

u must be lacking information.
$5000 for expenses deducted from the capital is not evidence of how much the capital was.
All you told us was each year he adds 1/3 of his current capital to his capital, and that after 3 years $15 000 worth of EXPENSES were deducted.

that means nothing regarding the amount of Capital he had or gained.

C (the amount of capital at the beginning) grows to:

C* 4/3 -5000 at the end of year 1

[(C * 4/3 -5000) * 4/3)] -5000 at the end of year 2 and

{[(C * 4/3 -5000) * 4/3)] -5000}*4/3 -5000 at the end of year 3 = 2C

Then solve for C, to obtain C = 55500

That was the complete problem.

Now, I understand your first solution more than your second, but I am not totally clear.