Geometric Sequence Calculator
nth term, sum, and whether the infinite series converges.
Work out Geometric Sequence. nth term, sum, and whether the infinite series converges. Shows the working, not just the answer.
Term 5
162
sum of 5 terms is 242 — diverges
The nth term is ar^(n−1) and the sum of n terms is a(1−rⁿ)/(1−r). With a ratio of 1 or more the terms do not shrink, so there is no infinite sum. The formula a/(1−r) will still return a number for r = 2 — a negative one — and it means nothing. This page reports no infinite sum rather than that figure. A ratio of exactly 1 makes every term identical, and the sum is simply n times the first term — handled here without dividing by zero.
How the Geometric Sequence Calculator works
Enter a first term, a common ratio and a term count. The infinite sum is reported only when the ratio is below 1 in magnitude — the formula returns a number for larger ratios too, and that number is meaningless.
Also known as: geometric series sum calculator · common ratio calculator · infinite geometric series calculator · nth term geometric sequence
Frequently asked questions
What is the nth term of a geometric sequence?
ar^(n−1), where a is the first term and r the common ratio. As with arithmetic sequences the exponent is n−1, not n.
When does an infinite geometric series converge?
When |r| < 1, so the terms shrink towards zero fast enough. Then the sum is a/(1−r). At |r| ≥ 1 the terms do not shrink and the series grows without bound.
Why does 1 + ½ + ¼ + ⅛ … equal exactly 2?
With a = 1 and r = ½ the formula gives 1/(1−½) = 2. Each term covers half the remaining gap to 2, so the total approaches it without ever exceeding it.
What if the ratio is exactly 1?
Every term is identical and the sum is simply n times the first term. The standard formula divides by 1−r and would fail, which is why that case is handled separately.
Put this calculator on your own site
Free to use, on any site, commercial or not. Paste this where you want it to appear. It is a plain iframe, so it works in WordPress, Squarespace, Wix, Webflow, Ghost and anything else that accepts HTML.
<iframe src="https://www.thecalclibrary.com/embed/geometric-sequence-calculator" width="100%" height="640" style="border:1px solid #e2e8f0;border-radius:12px" loading="lazy" title="Geometric Sequence Calculator"></iframe>
<p style="font:13px/1.5 system-ui,sans-serif;margin:6px 0 0;color:#64748b">Powered by <a href="https://www.thecalclibrary.com/geometric-sequence-calculator" style="color:#64748b">Geometric Sequence Calculator</a> from The Calc Library</p>The only condition is that the credit line below the frame stays in place. That one line is what pays for the tool being free — it is how anyone else finds it.