Growth and Decay

Sometimes the same percentage change repeats period after period — money earning interest every year, a population growing every season. When a change repeats, you don't add the percentage each time; you multiply by the same multiplier each time.

A r\% increase means multiplying by 1 + \tfrac{r}{100}. Doing that for n periods means multiplying by it n times — that is, raising it to the power n:

\text{amount} = P \times \left(1 + \tfrac{r}{100}\right)^{n}

This is compound growth (for money, compound interest). It grows faster than simple interest, because each period's interest is itself added to the total and then earns interest of its own.

When the quantity shrinks by the same percentage each period — a car depreciating, a substance halving — the multiplier is less than 1. A r\% decrease means multiplying by 1 - \tfrac{r}{100} each time:

\text{amount} = P \times \left(1 - \tfrac{r}{100}\right)^{n}

This is compound decay (or depreciation). The structure is identical to growth — only the multiplier flips from above 1 to below 1.

For a percentage change of r\% repeated over n periods, starting from P:

growth uses a multiplier greater than 1: 1 + \tfrac{r}{100};
decay uses a multiplier less than 1: 1 - \tfrac{r}{100};
applying the multiplier n times means raising it to the power n;
over time, compound interest beats simple interest, because past interest earns interest too.

Watch it compound

Here is 100 \times \left(1 + \tfrac{r}{100}\right)^{x} — £100 growing for x years. Pull the rate slider and notice how the curve bends upward: a steady percentage rate gives an ever-steepening climb, not a straight line.