Growth and Decay

Money sitting in a savings account. A colony of rabbits in a good summer. A radioactive sample in a lab. A cup of coffee going cold on the desk. These look like completely different things — but they all change in the same way: by a fixed percentage each step.

That single pattern is exponential growth and decay, and here's the headline: the numbers grow (or shrink) far faster than most people expect. Something creeping up by "just a few percent" a year can double, and double again, until it runs away with itself. The maths for all of it is the same, and it's surprisingly simple.

The method: repeated multiplication

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.

Worked example 1 — compound interest

You put \pounds 1000 in an account paying 5% a year. What is it worth after 3 years?

Each year multiplies by 1.05, and there are 3 years:

1000 \times 1.05^{3} = 1000 \times 1.157625 = \pounds 1157.63

Year by year it's £1050, then £1102.50, then £1157.63. Each year's 5% is bigger than the last, because it's 5% of a bigger pile — that's the compounding.

Worked example 2 — depreciation (decay)

A new car costs \pounds 12{,}000 and loses 15% of its value every year. What is it worth after 3 years?

Losing 15% keeps 85%, so multiply by 0.85 each year:

12{,}000 \times 0.85^{3} = 12{,}000 \times 0.614125 = \pounds 7369.50

Notice it does not lose 3 \times 15\% = 45\% (which would leave £6,600). Because each 15% comes off a smaller value, the car keeps a little more than the naïve guess — decay slows itself down.

Worked example 3 — compound vs simple interest

£1000 at 5% for 3 years. Compare the two ways interest can be paid:

Simple interest: 5% of the original £1000 is £50, paid three times → 1000 + 3 \times 50 = \pounds 1150.
Compound interest: 1000 \times 1.05^{3} = \pounds 1157.63.

Compound pulls ahead by £7.63 after just 3 years — small now, but the gap widens every year, because compound interest keeps earning interest on its own interest.

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.

Watch the £1000 grow, year by year

Compounding is easiest to believe when you see it step by step. Here is that same \pounds 1000 at 5%, with the interest earned each year in the last column — notice it grows every year, because each 5% is taken on a bigger balance:

Start: \pounds 1000.00
After year 1: 1000 \times 1.05 = \pounds 1050.00 (earned £50.00)
After year 2: 1050 \times 1.05 = \pounds 1102.50 (earned £52.50)
After year 3: 1102.50 \times 1.05 = \pounds 1157.63 (earned £55.13)

Simple interest would earn a flat £50 every year forever. Compound interest earns a little more each year — £50, then £52.50, then £55.13 — and that gently rising gap is the whole reason 1.05^{n} eventually leaves a straight line far behind.

Exponential growth and decay is repeated multiplication, not repeated addition. Each period's change is worked out on the new amount, not the starting amount.

So "5% per year for 3 years" is 1.05^{3} \approx 1.158 — a 15.8% total increase — not 3 \times 5\% = 15\%. The difference is tiny at first (0.8% here), but it grows and grows: over 20 years, 5% compound turns £100 into about £265, while "15% × 4 = 60% then add it on" would only ever give the wrong answer by more and more. This runaway is exactly why compound interest is so powerful — and why it never pays to reach for n \times r\%.

A legend tells of an inventor who asked a king for a "modest" reward: one grain of rice on the first square of a chessboard, two on the second, four on the third — doubling each square. The king laughed and agreed. By square 64 he owed more rice than exists on all of Earth — over 18 quintillion grains. That's doubling, which is just repeated multiplication by 2: pure exponential growth.