By · Updated

Geometric Sequence Calculator

Common Ratio, Sum to Infinity and nth Term, With the Working for Each

No ratings yet
Result

—

OneTask is free, with no sign-up, and built by one person.
If this tool saved you time, a coffee helps keep new ones coming.

How the Geometric Sequence Calculator works

This geometric sequence calculator gives the nth term, the sum of the first n terms and, when the ratio is between −1 and 1, the sum to infinity. Enter the first term and the common ratio, the number every term is multiplied by to get the next. The working for each formula is written out, and the first 24 terms are listed with running totals.

A geometric sequence multiplies by the same number each time. 3, 6, 12, 24 doubles at every step, so its common ratio is 2. A ratio above 1 makes the terms grow fast, a ratio between 0 and 1 makes them shrink towards 0, and a negative ratio makes the signs flip between plus and minus.

How the geometric sequence calculator works

The nth term formula is aₙ = a₁ × rⁿ⁻¹. The sum of the first n terms is Sₙ = a₁(1 − rⁿ) ÷ (1 − r), or simply n × a₁ when r is 1. When r is between −1 and 1 the terms shrink to nothing and the sum settles on a limit, S∞ = a₁ ÷ (1 − r).

First example. For 3, 6, 12, … the 10th term is 3 × 2⁹ = 1,536 and the first ten terms add up to 3 × (1 − 2¹⁰) ÷ (−1) = 3,069.

Second example. For 16, 8, 4, … the ratio is 0.5, so the terms halve. The sum of all of them, forever, is 16 ÷ (1 − 0.5) = 32.

How many grains of rice are on the 64th square of a chessboard, if the first has 1 and each square doubles? The 64th square holds 2⁶³, about 9.2 × 10¹⁸ grains, and the whole board holds 2⁶⁴ − 1.

The first 24 terms of 3, 6, 12, … with running totals

n Term aₙ Sum Sₙ
1 3 3
2 6 9
3 12 21
4 24 45
5 48 93
6 96 189
7 192 381
8 384 765
9 768 1,533
10 1,536 3,069
11 3,072 6,141
12 6,144 12,285
13 12,288 24,573
14 24,576 49,149
15 49,152 98,301
16 98,304 196,605
17 196,608 393,213
18 393,216 786,429
19 786,432 1,572,861
20 1,572,864 3,145,725
21 3,145,728 6,291,453
22 6,291,456 12,582,909
23 12,582,912 25,165,821
24 25,165,824 50,331,645

Tips

FAQ

What is the difference between arithmetic and geometric sequences?

An arithmetic sequence adds the same amount each time, such as 2, 5, 8, 11. A geometric sequence multiplies by the same amount, such as 2, 6, 18, 54. Over many terms the geometric one wins by a huge margin whenever its ratio is above 1, because the growth compounds.

How can an infinite sum be a finite number?

When each term is a fixed fraction of the one before, the terms get small so quickly that the total stops growing in any meaningful way. 1 + 1/2 + 1/4 + 1/8 + … gets closer and closer to 2 but never passes it, so 2 is the sum to infinity.

Why does 0.999 recurring equal 1?

0.999… is the geometric series 0.9 + 0.09 + 0.009 + …, with a₁ = 0.9 and r = 0.1. Its sum to infinity is 0.9 ÷ (1 − 0.1) = 0.9 ÷ 0.9 = 1, exactly. Enter 0.9 and 0.1 to see the running totals close in on 1.

What happens with a negative ratio?

The signs alternate. With a₁ = 10 and r = −0.5 the sequence is 10, −5, 2.5, −1.25, … and the sums swing above and below their limit, 10 ÷ 1.5 ≈ 6.667, getting closer each time.

See also: all Trigonometry and Calculus tools, Arithmetic, Number Theory, Algebra and Geometry, Statistics.

Related tools

Arithmetic Sequence Calculator

The nth term, the sum and the rule, with the working.

Summation Calculator

Sigma notation sums with the terms written out.

Exponential Growth And Decay Calculator

Final amount, doubling time or half-life for steady percentage growth or decay.

Compound Interest Calculator

Growth on a balance, with regular contributions and the real rate.

Definite Integral Calculator

The integral between two limits, the area and the average value.