Math Playground
Money

Compound interest derivation

Why A = P(1 + r/n)^(nt) — and where every letter comes from.

If P is the principal, r is the annual rate, n is the number of times compounded per year, and t is years:

Walk through
Step 1 of 5
Start

You deposit principal **P** at annual rate **r**, compounded **n** times a year.

Try it

$2,000 at 6% compounded monthly for 3 years

A = 2000(1 + 0.06/12)^(12·3) = 2000(1.005)³⁶ ≈ $2,393.46.

More frequent compounding helps, but with diminishing returns: at 6%, going from annual to monthly adds value; monthly to continuous barely moves the needle.

Your turn

If r/n = 0.01 and there are 24 periods, by what factor does the money grow?

Compound interest

Why?

Each compounding period multiplies by (1 + r/n). After n·t periods, you've multiplied that many times.

As n → ∞ (continuous compounding), A → P·e^(rt).