What Is a Geometric Sequence?

You have a list of numbers where each term after the first is multiplied by the same fixed number. That is what a geometric sequence is. The multiplier is called the common ratio, symbolised as r. For the sequence 2, 6, 18, 54, you multiply by 3 each time, so the first term a₁ = 2 and r = 3. This is the geometric sequence definition you will use for everything that follows: a constant non-zero ratio between consecutive terms. Recognising one means checking that dividing any term by the one before it always gives the same number.

The notation matters. The first term is a₁. The term at position n is aₙ, given by the explicit formula aₙ = a₁ × r⁽ⁿ⁻¹⁾. The index starts at n = 1, and the domain is the positive integers. This formula is from OpenStax Algebra and Trigonometry 2e, Section 13.3, and it is how you find any term without listing all the ones before it.

To decide if a sequence is geometric, pick two consecutive terms, divide the later one by the earlier one, and write down the result. Then do it for the next pair. If the quotient is the same for every pair, the sequence is geometric. That constant quotient is the common ratio r. The formula is r = aₙ / aₙ₋₁.

For example, test 3, 9, 27, 81. 9 ÷ 3 = 3, 27 ÷ 9 = 3, 81 ÷ 27 = 3. The common ratio is 3. Now test 2, 4, 8, 16. 4 ÷ 2 = 2, 8 ÷ 4 = 2, 16 ÷ 8 = 2. Same pattern. Contrast with an arithmetic sequence, which adds a constant difference instead of multiplying by a constant ratio. The arithmetic vs geometric sequence distinction is the single thing newcomers get wrong most often: arithmetic uses addition, geometric uses multiplication.

One edge case catches students. If the terms are 5, 5, 5, 5, the ratio is 5 ÷ 5 = 1 every time. This is a geometric sequence with r = 1. It is a constant sequence, and it is still geometric. Another edge: 0, 0, 0, 0. The ratio is 0 ÷ 0, which is undefined. A geometric sequence cannot have a zero term after a non-zero first term because the common ratio would be undefined. The research from OpenStax implicitly requires the ratio to be non-zero. If the first term itself is zero, every term is zero, but the ratio is undefined; mathematicians generally exclude this case from the definition of a geometric progression.

The value of the common ratio r determines whether the geometric sequence grows, decays, alternates, or stays flat. The behaviour falls into five distinct ranges, and each one changes what you can do with the sequence and what the sum formulas will tell you.

When r > 1

The terms increase in magnitude without bound. Example: 1, 2, 4, 8, 16 with r = 2. The sequence diverges: there is no finite limit. The infinite geometric series sum formula does not apply because the terms never approach zero. This is the compound interest pattern and the population explosion case.

When 0 < r < 1

The terms get smaller and approach zero. Example: 100, 50, 25, 12.5 with r = 0.5. This is the depreciation or radioactive decay pattern. The infinite sum formula S∞ = a₁ / (1 − r) works here because |r| < 1, and the series converges to a finite number.

When r < 0

The terms alternate sign. For r = −2, the sequence 5, −10, 20, −40 alternates positive, negative, positive, negative. The magnitude grows because |r| > 1, so the series diverges. For r = −0.5, the sequence 8, −4, 2, −1, 0.5 alternates and shrinks toward zero. Because |r| < 1, the infinite sum converges. The alternating sign does not by itself determine convergence; only the absolute value matters.

When r = 1

Every term equals the first term. The sequence 3, 3, 3, 3 is geometric. The partial sum formula Sₙ = a₁(1 − rⁿ) / (1 − r) is undefined for r = 1 because you would divide by zero. Instead, use Sₙ = n × a₁. The infinite sum does not exist because the terms never approach zero.

When r = −1

The sequence bounces between two values: a₁, −a₁, a₁, −a₁. Example: 5, −5, 5, −5. The terms never settle to a single number. The sum of an even number of terms is zero; the sum of an odd number is a₁. The infinite sum does not converge.

Range of r	Term Behaviour	Infinite Sum Exists?	Partial Sum Formula
r > 1	Grows without bound	No (diverges)	a₁(1 − rⁿ) / (1 − r)
0 < r < 1	Decays toward zero	Yes (converges to a₁/(1−r))	a₁(1 − rⁿ) / (1 − r)
r < 0, |r| > 1	Alternates, grows in magnitude	No (diverges)	a₁(1 − rⁿ) / (1 − r)
r < 0, 0 < |r| < 1	Alternates, decays to zero	Yes (converges to a₁/(1−r))	a₁(1 − rⁿ) / (1 − r)
r = 1	Constant sequence	No (terms do not approach zero)	Sₙ = n × a₁
r = −1	Alternates between a₁ and −a₁	No (does not settle)	Sₙ = 0 for even n, a₁ for odd n

Geometric sequences model situations where something changes by a constant percentage each period. Here are four common applications, with the numbers you would actually use.

Compound Interest

Deposit $1,000 at 5% annual interest compounded yearly. The balance after each year is a geometric sequence. The common ratio is 1.05. After year 1, the balance is $1,050. After year 2, it is $1,102.50. After year 3, it is $1,157.63. The explicit formula gives the balance at any year directly. OpenStax Algebra and Trigonometry 2e covers this in Section 13.4 as the standard application of a finite geometric series.

Depreciation

A car worth $30,000 loses 20% of its value each year. The common ratio is 0.80. After year 1, the car is worth $24,000. After year 2, $19,200. After year 5, $9,830.40. This is a convergent sequence because 0 < r < 1, and the value approaches zero over many years. The finite sum formula would tell you the total value lost over a period.

Population Growth

A town of 5,000 people grows at 3% per year. The common ratio is 1.03.This is the same structure as compound interest but applied to people. The sequence diverges because r > 1, meaning the population grows without bound in the model, real-world constraints eventually break the pattern.

Halving and Radioactive Decay

A 200-gram radioactive sample has a half-life of 1 year. Each year, the remaining mass is half of the previous year. The common ratio is 0.5. After year 1, 100 g remain. After year 2, 50 g. After year 5, 6.25 g. The sequence converges to zero.This is the classic example of a convergent geometric series.

You will often need to decide whether a given list of numbers is a geometric progression. There are two reliable ways to do it.

The Division Test

Take any term and divide it by the term before it. Repeat for every adjacent pair. If the quotient is the same each time, the sequence is geometric. This works even if the numbers are fractions or decimals. For the sequence 32, 16, 8, 4, divide: 16 ÷ 32 = 0.5, 8 ÷ 16 = 0.5, 4 ÷ 8 = 0.5. The common ratio is 0.5, and the sequence is geometric.

Finding r from Two Non-Consecutive Terms

If you are given aₘ and aₙ, the formula is r = (aₙ / aₘ)^(1/(n-m)). Suppose you know term 3 is 12 and term 5 is 48. The difference in indices is 2. Compute (48 / 12)^(1/2) = 4^(1/2) = 2. The common ratio is 2. Now you can find the first term: a₁ = 12 / 2² = 3. The sequence is 3, 6, 12, 24, 48. Watch the sign. If the ratio of the terms is negative and the root index is even, you get a non-real result. For example, a₂ = 8 and a₄ = −32 gives (−32 / 8)^(1/2) = (−4)^(1/2), which is not a real number. In that case, r is −2, and you must deduce the sign by considering whether the sequence alternates.