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calculating future value: initial deposit and recurring deposits of a fixed but different value
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but how do i bring the initial deposit into the equation? basically, you can't.  unless you combine two different formulas from math of finance into a single expression. the single initial deposit of $1000 will compound for 20 years at 5% compounded annually. the final amount for this part of the deposit will be: v1 = 1000 x (1.05)^20 in addition the series of 20 payments will be an ordinary annuity with a regular payment of $100, with the value on the occasion of the 20th payment given by:  so the final total amount in the account at the end of 20 years will be the sum of these two values...
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