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To find a formula for future value, we'll write P for your starting principal, and r for the rate of return expressed as a decimal.
(So if the interest rate is 5%, r equals .05).
Your balance will grow according to the following schedule:
| Year | Balance |
| Now | P |
| 1 | P + rP |
| 2 | (P + rP) + r(P + rP) |
This starts to get messy in a hurry.
But you can simplify it by noticing that you can keep pulling out factors of (1 + r) from each line.
If you do that, the balances collapse to a simple pattern:
| Year | Balance |
| Now | P |
| 1 | P(1 + r) |
| 2 | P(1 + r)2 |
If you follow this pattern out for n years, you get the general formula for future value:
(Also see more compounding to compound interest periodically or continuously,
and basic investment for compound interest with annual additions.)
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