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Following scenario: I open a savings account and I deposit 1,000 USD. Now, every year I deposit another 100 USD. The interest rate is 5%. I want to know how much money will be on that savings account in 20 years.

I understand that if the initial deposit had been 100 as well you could just do: 100 * 1/0.05 * (((1+0.05)^20)-1) or C * 1/r * (((1+3)^N)-1)

But how do I bring the initial deposit into the equation?

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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:

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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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If I is the initial deposit, P the periodic deposit, r the rent per period, n the number of periods, and F the final value, than we can combine two formulas into one to get the following answer:

F = I*(1+r)n + P*[(1+r)n-1]/r

In this case, you get V = 1000*(1.05)20 + 100*[(1.05)20-1]/0.05 = 5959.89 USD.

Note that the actual final value may be lower because of rounding errors.

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Illustrating with a shorter example: Suppose I deposit 1,000 USD. Every year I deposit another 100 USD. I want to know how much money will be on that savings account in 4 years.

enter image description here

The long-hand calculation is

1000 (1 + 0.05)^4 + 100 (1 + 0.05)^3 +
 100 (1 + 0.05)^2 +  100 (1 + 0.05)^1 + 100 (1 + 0.05)^0 = 1646.52

Expressed with a summation

enter image description here

And using the formula derived from the summation (as shown by DJohnM)

1000 (1 + 0.05)^4 + 100 ((1 + 0.05)^4 - 1)/0.05 = 1646.52

So for 20 years

1000 (1 + 0.05)^20 + 100 ((1 + 0.05)^20 - 1)/0.05 = 5959.89

Note in year 20 (or year 4 in the shorter example) the final $100 deposit does not have any time to accrue interest before the valuation of the account.

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