How to Find the nth Term and Sum of a Sequence
To find the nth term of a geometric sequence use aₙ = a₁ × r^(n − 1), and for the sum of the first n terms use Sₙ = a₁ × (1 − rⁿ) ÷ (1 − r) — the ByteTools Geometric Sequence Calculator does both instantly from three inputs. Enter the first term, the common ratio and n, and it returns the nth term, the finite sum, the opening terms and, when the series converges, the infinite sum.
A geometric sequence multiplies each term by a fixed ratio — 2, 6, 18, 54 has a ratio of 3. This guide shows how to get every result the calculator offers.
What you need before you start
Three values fully describe a geometric sequence:
- First term (a₁) — where the sequence begins.
- Common ratio (r) — the number each term is multiplied by. It can be negative or fractional.
- n — how many terms you care about.
Step by step
- Enter the first term. For 5, 10, 20, 40 that is 5.
- Enter the common ratio. Divide any term by the one before it; here 10 ÷ 5 = 2.
- Enter n. Say you want the 6th term and the sum of the first 6 terms, so n = 6.
- Read the nth term and finite sum. The tool applies both formulas and lists the opening terms so you can sanity-check.
- Note the infinite sum when the ratio sits between −1 and 1, then click Copy to grab the results.
The formulas it uses
| Quantity | Formula | Condition |
|---|---|---|
| nth term | aₙ = a₁ × r^(n − 1) | Any r |
| Sum of n terms | Sₙ = a₁ × (1 − rⁿ) ÷ (1 − r) | r ≠ 1 |
| Sum of n terms | Sₙ = n × a₁ | r = 1 |
| Infinite sum | S = a₁ ÷ (1 − r) | |r| < 1 |
A worked example
Take a₁ = 3, r = 2, n = 5. The 5th term is 3 × 2⁴ = 48. The sum of the first five terms is 3 × (1 − 32) ÷ (1 − 2) = 93. The calculator shows both, plus the list 3, 6, 12, 24, 48 so you can verify at a glance. Because the ratio is 2, there is no infinite sum — the terms grow without bound.
Everything runs in your browser with JavaScript, so it is instant, private and works offline once loaded.
Try the Geometric Sequence Calculator — free and 100% in your browser.
FAQ
How do I find the common ratio if I only have the terms?
Divide any term by the previous one. If the result is the same for every pair, that is your common ratio. Enter it and the calculator handles the rest.
Can n be a large number?
Yes. The formulas use exponentiation rather than looping, so even a large n returns instantly. Very large ratios raised to a big power produce very large numbers, which is expected.
Do I need an internet connection?
No. Once the page has loaded, the math runs locally in your browser, so it works offline and nothing you enter is uploaded.
What if the ratio is a fraction?
Fractional ratios are fine. If the fraction's magnitude is below 1, the tool also reports the infinite sum, since the series converges.
Related free tools
- Arithmetic Sequence Calculator — for sequences that add a constant instead of multiplying.
- Exponent Calculator — evaluate the powers behind the nth term.
- Logarithm Calculator — solve for n when you know the term.
- Fibonacci Calculator — explore another famous sequence.
Built by ByteVancer
ByteTools is a free product of ByteVancer, a software and web development studio building web apps, SaaS and custom software. If fast, private tools like this appeal to you, explore what ByteVancer can build for you.
Recommended reading
Geometric Sequence: Tips and Common Mistakes
Avoid the classic geometric sequence errors: off-by-one nth-term exponents, mixing up ratio direction, and misjudging when a series converges.
Geometric Sequences in the Real World: Examples
Where geometric sequences show up in real life: compound growth, depreciation, bouncing balls, and doubling patterns, with worked examples.
How to Find the nth Term and Sum of a Sequence
Step-by-step guide to finding the nth term and sum of an arithmetic sequence with a free, private, in-browser calculator.
Yes or No Generator: Real Use Cases and Examples
From beating decision paralysis to games and classrooms, see real use cases and examples for a random yes or no generator.