Math calculator
Terms that multiply — and whether they add up to anything finite.
Exact ratios, including fractions.
aₙ = 2 × 3⁽ⁿ⁻¹⁾
39366
The 10th term. The exponent is n − 1, so this is 2 multiplied by the ratio 9 times — the sequence grows without bound.
Term 10
39366
from the closed form
Sum of first 10
59048
the partial sum
Infinite sum
none
the terms do not shrink
Recursive form
a₁ = 2, aₙ = aₙ₋₁ × 3
the same rule, step by step
The first 10 terms
The formulas, with your numbers in
| What | Formula | Value |
|---|---|---|
| nth term | aₙ = 2 × 3⁽ⁿ⁻¹⁾ | 39366 |
| Sum to n | Sₙ = 2 × (1 − 3ⁿ) ÷ (1 − 3) | 59048 |
| Sum to infinity | does not exist | — |
Ratios are kept as fractions, so a ratio of 1/3 stays exact instead of becoming 0.3333333333.
1 + ½ + ¼ + ⅛ + … never stops, and adds up to exactly 2. 1 + 2 + 4 + 8 + … adds up to nothing at all. The difference is whether the ratio is smaller than 1, and this page says which side yours is on.
Exact ratios, including fractions.
Updated 7 September 2026 · Works in any browser, no installation
an = a1 × r(n−1). Multiply the first term by the ratio one fewer time than the term number. For 2, 6, 18, … term 10 is 2 × 3⁹ = 39,366. Whether the infinite series adds up to anything depends entirely on whether |r| is below 1.
At a glance
The first term has been multiplied by the ratio zero times. r⁰ is 1, so a1 × r⁰ is just a1, which is what you want.
Term 2 has one multiplication, term 3 has two. By term 10 there have been nine. Using n instead of n − 1 makes every answer a whole factor too large — with a ratio of 3, three times too large.
It is the same off-by-one as in arithmetic sequences, and it comes from the same place: terms are counted from 1, operations from 0.
Add infinitely many positive numbers and you might expect infinity. Sometimes you get 2.
1 + ½ + ¼ + ⅛ + … approaches 2 and never passes it. Each term covers half the remaining gap, so the gap shrinks towards nothing without the total ever exceeding 2. The infinite sum is exactly 2.
The condition is |r| < 1: the terms have to shrink, and shrink fast enough. Then S = a1 / (1 − r), and for a1= 1 with r = ½ that is 1 / (1 − ½) = 2.
If |r| is 1 or more, the terms do not shrink and the partial sums run away. There is no infinite sum — not a very large one, none. The page reports that rather than printing the partial sum and leaving you to assume.
r = 1. Every term is the same, so the sum is n × a1. The usual formula divides by 1 − r and fails; the page uses the right one.
r = 0. Everything after the first term is zero, and the sum is just a1. A degenerate case, and a real one.
r = −1. The sequence alternates 1, −1, 1, −1, … and the partial sums bounce between 1 and 0 forever without settling. This is Grandi’s series, and it is why “1 − 1 + 1 − 1 + …” has no sum in the ordinary sense however tempting ½ looks.
Negative ratios generally. Terms alternate in sign. They still converge whenever |r| < 1, so r = −½ gives a perfectly good total — convergence cares about size, not sign.
Money growing at 7% a year is a geometric sequence with ratio 1.07. After n years the balance is the original times 1.07n.
That is the whole difference between compound and simple interest. Simple interest adds a fixed amount each year and is arithmetic; compound multiplies and is geometric. Over a couple of years they are close. Over thirty they are not remotely.
The same shape covers population growth, radioactive decay with a ratio below 1, and anything described as growing “by a percentage” rather than “by an amount”. The word to listen for is per cent: it means the change is proportional, and proportional change is geometric.
Look at what stays the same between consecutive terms.
Constant difference → arithmetic. 3, 8, 13, 18: subtract and you always get 5. These grow in a straight line.
Constant ratio → geometric. 3, 6, 12, 24: divide and you always get 2. These curve, and eventually they curve hard.
Any geometric sequence with a ratio above 1 will eventually overtake any arithmetic sequence, whatever the head start. That single fact is behind compound interest, viral spread, and most of the surprises in both.
If you have the terms but do not know which family they belong to, the Nth Term Calculator tests both and reports which fits.
Not approximately. Exactly. And a geometric series is the shortest honest proof.
0.999… is 9/10 + 9/100 + 9/1000 + …, which is geometric with a1= 9/10 and r = 1/10. Since |r| < 1, the infinite sum is a1/(1 − r) = (9/10)/(9/10) = 1.
You can check it on this page: enter 9/10 as the first term and 1/10 as the ratio, and the infinite sum comes back as 1 exactly, because the arithmetic is done on fractions rather than decimals.
The same machinery turns any repeating decimal into a fraction. 0.333… is (3/10)/(1 − 1/10) = 1/3, which is where the familiar result comes from.
The convergence condition is a theorem rather than a convention; these are the references.
Method. Terms and sums are carried as exact rationals, so a ratio of 1/3 stays a third rather than becoming a repeating decimal, and the partial sum comes from the closed form rather than a loop. The convergence test compares |r| with 1 exactly, on integers, so a ratio of 99/100 is correctly reported as convergent rather than falling foul of a floating-point comparison. Where the terms would grow past what can be displayed, the page says so rather than truncating silently. That engine is verified on every change against 76 hand-written assertions, including that the closed-form sum matches a term-by-term loop on six hundred generated sequences, and that a ratio of exactly 1 or −1 is refused an infinite sum with the reason. The count and the per-case breakdown are published on the formula verification page.
If the terms step by a fixed amount rather than a fixed factor, the Arithmetic Sequence Calculator applies. To average a set of growth rates rather than project one, the Geometric Mean Calculator is the right tool.
Educational use disclaimer
This is an educational tool. The arithmetic is exact for whatever you enter; whether a real quantity really multiplies by a fixed factor each period is a modelling question, and compound growth rarely continues indefinitely.
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Published 7 September 2026