Why does repeated percentage change compound?

Repeated percentage change compounds because each change acts on the amount present at that stage. A percentage is a proportion of a base, and the base is updated after every interval. A constant percentage therefore gives a constant multiplier, not a constant number of units added.

The invented amount begins at 100 units and increases by 20% per interval. Multiplying by 1.20 gives 120 after the first interval. The next 20% is taken from 120, so it adds 24 rather than 20. The second amount is 144. Adding a fixed 20 units each interval would instead produce 100, 120 and 140.

For a constant signed decimal rate r, the multiplier is 1 + r. After n intervals the model is Aₙ = A₀(1 + r)ⁿ, where A₀ is the initial amount. The exponent records repeated multiplication. A 20% decrease uses the factor 0.80, so two such decreases take 100 to 80 and then 64.

Equal percentage increases and decreases do not generally cancel. A 20% increase takes 100 to 120, but a subsequent 20% decrease removes 24 from that updated base. The result is 96, because 1.20 × 0.80 equals 0.96 rather than one.

These exact rates and isolated intervals define the mathematical example. Real processes can have changing rates, outside additions and limits, so the sequence is not a forecast merely because its formula is exponential.