Geometric sequences — Mathematics, 14–17 years
A sequence in which each term is made by multiplying the previous one by the same number. Mathematics, 14–17 years.
The same multiplier each time
In a geometric sequence, you move from one term to the next by multiplying by a fixed number, called the common ratio. For example, 3, 6, 12, 24 uses a ratio of 2; the terms do not grow by equal additions, but by equal scaling.
Why use a multiplier?
Many changes are proportional: a price rises by 5%, a population loses 10%, or money earns interest. Adding a fixed amount cannot describe these changes, because the change depends on the current amount. A geometric sequence keeps the same percentage or scale from one step to the next.
A growing savings balance
You deposit €200 and the account grows by 3% each month. The multiplier is 1.03. After one month: 200 × 1.03 = €206; after two: 206 × 1.03 = €212.18; after three: 212.18 × 1.03 = €218.55, rounded to cents. The same result can be written 200 × 1.03³.
Confusing percentage with points
A common mistake is to treat a 20% increase as adding 20 each time. That is reasonable when the first amount is 100, because 20% of 100 is 20. But 20% of 150 is 30, so the correct multiplier is 1.20, not “add 20”; the base changes after every step.
Where it appears
Geometric sequences model repeated percentage change: compound interest, depreciation of a machine, drug concentration falling in equal fractions, and the spread of a population under a simple model. They are useful when each period acts on the result of the previous period, not when a fixed amount is added.
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