Geometric Sequence Calculator

What Is a Geometric Sequence?

A geometric sequence (or geometric progression) is a list of numbers in which each term is the previous term multiplied by a fixed number, the common ratio r. In 2, 6, 18, 54, … each term is 3 times the one before, so r = 3. In 64, 32, 16, 8, … each term is half the previous one, so r = 1/2.

Common Ratio

Divide any term by the one before it: r = aₙ/aₙ₋₁. For 5, 15, 45, 135: 15/5 = 3, 45/15 = 3 and 135/45 = 3, so r = 3. Check every pair: 2, 6, 12, 24 has ratios 3, 2 and 2, so it is not geometric, and averaging the ratios would be wrong.

Zero needs care. A ratio cannot be computed from 0/0, so 0, 0, 0, … does not determine r. In 5, 0, 0, … the first ratio 0/5 gives r = 0, and a sequence such as 0, 0, 5 cannot be geometric at all, because 0 × r is always 0.

Geometric Sequence Formula

The nth term is aₙ = a₁ · rⁿ⁻¹. To reach term n from term 1 you multiply by r once for each of the n − 1 steps. For 2, 6, 18, …: aₙ = 2 · 3ⁿ⁻¹, so a₆ = 2 · 3⁵ = 2 · 243 = 486.

How to Find the nth Term

Identify a₁ and r, then substitute n. For 3, 6, 12, 24, …: a₁ = 3 and r = 2, so a₇ = 3 · 2⁶ = 3 · 64 = 192. For 64, 32, 16, 8, …: a₇ = 64 · (1/2)⁶ = 64/64 = 1.

With a negative ratio, watch the sign: for 2, −4, 8, −16, … r = −2, so a₈ = 2 · (−2)⁷ = −256. Odd powers of a negative ratio are negative and even powers are positive.

How to Find the Sum of a Geometric Sequence

The sum of the first n terms is Sₙ = a₁(1 − rⁿ)/(1 − r), or equivalently a₁(rⁿ − 1)/(r − 1), which is easier when r > 1. For 3 + 6 + 12 + 24 + 48 + 96: S₆ = 3(2⁶ − 1)/(2 − 1) = 3 · 63 = 189.

The formula cannot be used when r = 1, because it would divide by 0. Then every term equals a₁ and Sₙ = n · a₁. When r = 0 only the first term is nonzero, so Sₙ = a₁.

Infinite Geometric Series

An infinite geometric series a₁ + a₁r + a₁r² + … converges only when |r| < 1. Then rⁿ → 0 and the partial sums approach S∞ = a₁/(1 − r). For 10 + 5 + 2.5 + 1.25 + …, r = 1/2 and S∞ = 10/(1 − 1/2) = 20. A negative ratio also works: with a₁ = 8 and r = −1/2, S∞ = 8/(3/2) = 16/3.

If |r| ≥ 1 the series diverges and has no sum. With r = 2 the terms grow without bound; with r = 1 the sum keeps adding a₁; with r = −1 the partial sums keep jumping between a₁ and 0. Using a₁/(1 − r) in these cases gives a meaningless number.

Geometric Mean

The geometric mean of two positive numbers a and b is G = √(ab). It is the middle term of the geometric sequence a, G, b: the geometric mean of 4 and 16 is √64 = 8, and 4, 8, 16 has r = 2.

To insert k geometric means between a and b, solve b = a · r^(k+1) for r and multiply by r repeatedly. Two means between 2 and 54 give r³ = 27, r = 3 and the sequence 2, 6, 18, 54. When k + 1 is even, −r works too; when b/a is negative and k + 1 is even, no real means exist.

How to Find a Missing Term

Use two known terms. If they are next to each other, divide to get r. If they are m positions apart, their ratio is rᵐ: in 81, __, 9, 3 the terms 81 and 9 are two apart, so r² = 1/9 and r = ±1/3; the term 3 = 9 · r fixes r = 1/3, so the missing term is 27.

If only two terms are known and they are an even number of positions apart, both signs of r can work. In 2, __, 18 the missing term could be 6 or −6, and a good answer says so.

Geometric Sequence vs Arithmetic Sequence

An arithmetic sequence adds the same number each time: 2, 5, 8, 11 has common difference d = 3. A geometric sequence multiplies by the same number: 2, 6, 18, 54 has common ratio r = 3.

Arithmetic sequences grow linearly, so their graph of n against aₙ is a straight line. Geometric sequences grow or decay exponentially. To tell which you have, check whether the differences or the ratios of consecutive terms are constant.

Geometric Sequence vs Geometric Series

The sequence is the list of terms: 2, 6, 18, 54, … The series is the sum of those terms: 2 + 6 + 18 + 54 + … A finite series adds the first n terms (Sₙ); an infinite series adds all of them and only has a value (S∞) when |r| < 1. “Find the 5th term” is a sequence question; “find the total of the first 5 terms” is a series question.

Growth and Decay

For a positive first term: r > 1 gives growth (2, 6, 18, …); 0 < r < 1 gives decay toward 0 (64, 32, 16, …); r = 1 gives a constant sequence; r = −1 flips between a₁ and −a₁; −1 < r < 0 shrinks toward 0 while alternating in sign; and r < −1 grows in size while alternating. A negative first term flips every sign but not the size pattern.

Percentage growth is geometric: growing 5% per year means multiplying by r = 1.05, and losing 20% per year means r = 0.8.

Worked Examples

2, 6, 18, 54, …: r = 3 and a₆ = 2 · 3⁵ = 486.

5, 15, 45, 135: r = 15/5 = 3, confirmed by 45/15 and 135/45.

3 + 6 + 12 + 24 + 48: S₅ = 3(2⁵ − 1)/(2 − 1) = 93.

10 + 5 + 2.5 + 1.25 + …: |r| = 1/2 < 1, so it converges to 10/(1 − 1/2) = 20.

2, −4, 8, −16, …: r = −2 and a₈ = 2(−2)⁷ = −256.

64, 32, 16, 8, …: r = 1/2 and a₇ = 64(1/2)⁶ = 1.

2, 4, 8, 15: the ratios are 2, 2 and 15/8, so it is not geometric.

Common Mistakes

Using rⁿ instead of rⁿ⁻¹ in the nth-term formula. Dividing in the wrong order (a₁/a₂ instead of a₂/a₁), which inverts r. Losing the sign of a negative ratio, for example writing −2⁷ when you mean (−2)⁷. Using Sₙ = a₁(1 − rⁿ)/(1 − r) when r = 1. Giving an infinite sum when |r| ≥ 1. Averaging unequal ratios. Treating 0/0 as a ratio. Confusing the sequence (the terms) with the series (their sum).