Key takeaways
- aₙ = a × r^(n - 1), where a is the first term and r is the common ratio.
- The exponent is (n - 1), not n, since the first term needs r raised to the power 0.
- r > 1 grows fast; 0 < r < 1 shrinks toward zero; negative r alternates sign.
- Divide any term by the one before it to find r.
In this guide
What is a geometric sequence?
In a geometric sequence, each term is the previous term multiplied by a fixed number, called the common ratio. 2, 6, 18, 54 is geometric with a common ratio of 3. Because it multiplies rather than adds, it grows much faster than an arithmetic sequence.
Why does the nth term formula matter?
Multiplying by the same ratio again and again by hand is slow, and the numbers grow fast enough that mistakes creep in quickly. This formula skips straight to any term, however far along the sequence you need.
Starting with just 3 people and doubling the audience every round, round 6 alone brings in dozens of new viewers — without you multiplying through rounds 1 to 5 first.
The formula
| Symbol | What it means | Example |
|---|---|---|
| aₙ | The term you want, at position n | 96 |
| a | The first term | 3 |
| r | The common ratio between terms | 2 |
| n | The position (term number) you want | 6 |
How to find the nth term step by step
- 1Identify the first term, a, and the common ratio, r. Divide any term by the term before it to find r.
- 2Subtract 1 from n to get the exponent.
(n - 1). - 3Raise r to that power.
r^(n - 1). - 4Multiply by the first term.
aₙ = a × r^(n - 1).
Enter your own first term, ratio, and position below to find any term instantly.
Try it yourself
Pre-filled with the example — change any value.
Common mistakes to avoid
- Mixing up common ratio with common difference. A geometric sequence multiplies (r); an arithmetic one adds (d) — use the arithmetic sequence calculator for the latter.
- Forgetting a negative ratio flips signs. r = -2 makes terms alternate positive and negative.
- Forgetting a fractional ratio shrinks terms. r = 0.5 makes terms shrink toward 0 instead of growing.
Tips and tricks
- To add up every term in the sequence rather than find just one, use the geometric series sum calculator.
- Compare growth speed against an arithmetic sequence with similar starting numbers — geometric growth pulls ahead fast.
- A ratio close to 1 (like 1.02) grows slowly at first but still accelerates the further out you go.
Where you'll use it in real life
- Social media: viral shares, where each viewer passes content to several new people.
- Biology: population doubling under ideal conditions.
- Physics: a bouncing ball's height, shrinking by the same ratio each bounce.
- Finance: the value of a single compounding investment at a specific point in time — see the compound interest calculator.
Quick summary
The nth term of a geometric sequence is the first term multiplied by the common ratio raised to (n - 1). It captures how fast multiplying — rather than adding — makes a sequence grow. Try the calculator above with your own numbers.
Worked example
You share a video with 3 friends. Each round, everyone who has seen it shares it with 2 new people, so the number of new viewers doubles each round. How many new people see it in round 6?
- 01aₙ = a × r^(n-1)
- 02a6 = 3 × 2^5
- 03a6 = 3 × 32
- 04a6 = 96
These numbers are pre-filled in the calculator above.
Frequently asked questions
How do I find the nth term of a geometric sequence?
Use aₙ = a × r^(n - 1), where a is the first term, r is the common ratio, and n is the position you want.
How do I find the common ratio?
Divide any term by the term right before it. The result should be the same no matter which pair of consecutive terms you use.
What happens if the common ratio is a fraction?
The terms get smaller and smaller (decay) instead of growing, approaching zero as n increases.
What's the difference between arithmetic and geometric sequences?
Arithmetic sequences add a fixed number each step. Geometric sequences multiply by a fixed ratio each step, which is why they grow much faster.
