Mathématiques

What Is a Geometric Sequence?

Each term multiplies by the same number. The common ratio in 3, 6, 12, 24 and in 80, 40, 20, and how to reach the 10th term without listing the ones before.

Geometric sequence

A sequence in which each term is found by multiplying the previous one by the same fixed number, called the common ratio.

Aussi appelé: Geometric progression
Où les élèves le rencontrent: Grade 9 or 10 mathematics, usually straight after arithmetic sequences, and again inside every compound interest and depreciation question.

La réponse en bref
A geometric sequence multiplies by the same fixed number each time, called the common ratio. In 3, 6, 12, 24 the ratio is 2; in 80, 40, 20 it is 0.5. The nth term is arⁿ⁻¹, so the 10th term of 3, 6, 12, 24 is 3 × 2⁹ = 1536.

Un exemple

3, 6, 12, 24, … r = 2 → 10th term = 3 × 2⁹ = 1536

Divide any term by the one before it: 6 ÷ 3 = 2, 12 ÷ 6 = 2, 24 ÷ 12 = 2. The ratio is the same throughout, so the sequence is geometric with r = 2. To reach the tenth term, notice that the first term has been multiplied by 2 no times at all, the second once, the third twice — so the tenth has been multiplied nine times. That is 3 × 2⁹ = 3 × 512 = 1536, and the exponent being one less than the term number is the part most often got wrong.

Divide, do not subtract

The test for a geometric sequence is division, and it has to give the same answer at every step. In 3, 6, 12, 24 each division gives 2. In 80, 40, 20 each gives 0.5, and the sequence shrinks — still geometric, because the requirement is a constant multiplier, not a growing one.

Checking only one pair of terms is not enough. 2, 4, 8, 14 starts with two ratios of 2 and then breaks, and a student who stopped after the first division has answered a question about a sequence that does not exist.

The ratio can be negative, which produces the alternating pattern that looks alarming and behaves normally: 2, −6, 18, −54 has r = −3, and every division confirms it.

Reaching term 10 without writing terms 5 to 9

The nth term is arⁿ⁻¹, where a is the first term and r the common ratio. The exponent is n − 1 rather than n because the first term has not been multiplied by anything yet — it is simply where the sequence starts.

That off-by-one is the standard error, and it is worth checking on a term you can see. For 3, 6, 12, 24, the formula gives a 4th term of 3 × 2³ = 24, which matches the sequence. Now n = 10 gives 3 × 2⁹ = 1536, reached in one calculation.

For the shrinking sequence 80, 40, 20 the same formula gives an 8th term of 80 × 0.5⁷ = 0.625. The terms get smaller and smaller without ever reaching zero, which is the sequence version of what an exponential graph does as it approaches its asymptote.

The same idea wearing different clothes

Most students meet geometric sequences twice without noticing, because the second appearance is not called one. Compound interest is a geometric sequence: an amount growing by 10 per cent a year is being multiplied by 1.1 every year, so the balances form a sequence with r = 1.1. Depreciation is the same thing with r below 1 — an item losing 15 per cent of its value annually has r = 0.85.

Radioactive half-life works this way too, with r = 0.5 applied once per half-life period. Recognising the connection turns three separate topics into one, and it explains why the compound interest formula and the nth-term formula have the same shape: both are a starting value multiplied by a fixed number, raised to a power that counts how many times the multiplication has happened.

Questions fréquentes

How do you tell a geometric sequence from an arithmetic one?

Try both tests. Subtract consecutive terms — if the answers are all equal, it is arithmetic. Divide consecutive terms — if those are all equal, it is geometric. 3, 6, 9 is arithmetic with a difference of 3; 3, 6, 12 is geometric with a ratio of 2. The first two terms alone cannot tell them apart.

Can the common ratio be negative?

Yes, and it produces a sequence whose terms alternate between positive and negative. 2, −6, 18, −54 has a common ratio of −3, since each term is the previous one multiplied by −3. The alternating signs are the giveaway, and dividing consecutive terms confirms it in one step.

Can a geometric sequence get smaller?

Yes, whenever the ratio lies between −1 and 1. In 80, 40, 20, 10 the ratio is 0.5 and the terms halve each time, approaching zero without ever arriving. Sequences like this describe depreciation and half-life, and they are as geometric as the ones that grow.

Why is the formula arⁿ⁻¹ and not arⁿ?

Because the first term has not been multiplied by the ratio yet. Term 1 is a, term 2 is ar, term 3 is ar², so the exponent always runs one behind the term number. Testing the formula on a term you can already see is the quickest way to catch the slip before it costs a mark.

Dernière mise à jour 21 août 2026