An excel spreadsheet is very helpful for these types of problems.
Let's first assume that both deposits and withdrawls are made at the end of the year. (Deposits at the beginning of the year slightly change the answer, but not the methodology).
So B1 (balance at the end of year 1) = 1000.
B2 = 1000 + B1*(1.08) = B1*(1+1.08)
B3 = B1*(1+1.08+1.08^)
B10 = B1*(sum(1.08)^(n-1) n goes from 1 to 10
find B10. Now for the withdrawals.
B11 = B10*(1.08) - W
B12 = B11*(1.08) - W*1.06
= B10*(1.08)^2 - W*(1.08) - W*(1.06)
B13 = B10*(1.08)^3 - W*(1.08)^2 - W*(1.06)*(1.08) - W*(1.06)
B15 = B10*(1.08)^5 - W*[(1.08)^4 + (1.06)*(1.08)^3 + (1.06)^2*(1.08)^2 + (1.06)^3*(1.08) + 1.06^4]
solve for W such that B15=0.
You can do this with algebra or you could use Excel and plug in values and iterate until you find the right solution.
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