Geometric Sequence Calculator

What this calculator does

A geometric sequence is a list of numbers where you multiply by the same amount every time instead of adding: 2, 6, 18, 54, always times 3. This one builds that sequence for you.

Give it the first term, the multiplier, and how many terms you want, and it lists them out. It runs right here in the browser.

Using the calculator

1. Enter the first term (a1), where it starts.
2. Enter the common ratio (r), the number each term is multiplied by.
3. Enter the number of terms (n).
4. Pick a precision for decimals, then press Calculate.

It lists the terms of the sequence. The last one in the list is the nth term. Reset clears the boxes.

What a geometric sequence is

The defining feature is a constant ratio. You get from one term to the next by multiplying by the same number every time, called the common ratio, r. In 2, 6, 18, 54 the ratio is 3. In 80, 40, 20, 10 it is one half, so the sequence shrinks.

This is the shape of anything that grows or shrinks by a percentage rather than a fixed amount: money earning compound interest, a population doubling, a drug clearing the body, a bouncing ball losing the same fraction of height each bounce. Multiplying, not adding, is what makes it geometric.

Finding any term: the nth term formula

Rather than multiplying your way along term by term, you can jump straight to any term:

a_n = a1 × r^{(n − 1)}

where a1 is the first term, r is the common ratio, and n is the position. The power is (n − 1) for the same reason as before: the first term is already there, so you only multiply by r the remaining (n − 1) times to reach term n. So the 5th term of 2, 6, 18 is 2 × 3⁴ = 2 × 81 = 162.

Growth, decay, and flipping signs

The ratio r tells you the whole character of the sequence at a glance:

• r greater than 1: the terms grow, and they grow fast. This is the runaway behaviour behind the old story of doubling grains of rice on a chessboard, where the numbers become astronomical within a couple of rows. It is exponential growth.
• r between 0 and 1: the terms shrink toward zero without ever quite reaching it. This is decay, like a radioactive sample halving each half-life.
• r negative: the terms flip sign as they go, alternating positive and negative, while their size follows the same growth or shrink rule.

A worked example

Take a first term of 2, a common ratio of 3, and 5 terms.

1. Start at 2, then keep multiplying by 3: 2, 6, 18, 54, 162.
2. The 5th term by the formula: 2 × 3^{(5 − 1)} = 2 × 3⁴ = 2 × 81 = 162, which is the last term in the list.

The surprising bit: an endless sum that stops

Here is something that catches everyone the first time. If the ratio is between -1 and 1, so the terms keep shrinking, you can add up infinitely many of them and still get a finite total. Take 1/2 + 1/4 + 1/8 + 1/16 and keep going forever: the running total creeps closer and closer to exactly 1, and never passes it. An endless list of numbers with a definite sum. When you actually need the total of a geometric series, whether a few terms or the infinite case, the sum of series calculator handles it.

On decimals and precision

Multiplying decimals over and over is exactly where ordinary browser maths tends to accumulate small rounding errors. This tool sidesteps that by running the calculation through Decimal.js, a library made for exact decimal arithmetic, with the precision dropdown setting how many places it keeps. For clean whole-number ratios you will not notice, but for decimal ratios over many terms, more precision keeps the later terms honest.

Questions people ask

How do I find the nth term of a geometric sequence?

Use a_n = a1 × r^{(n − 1)}. Start from the first term and multiply by the common ratio (n − 1) times.

What is the common ratio?

The fixed number each term is multiplied by to get the next. Divide any term by the one before it to find it.

Can a geometric sequence get smaller?

Yes. A ratio between 0 and 1 makes each term a fraction of the one before, so the sequence shrinks toward zero.

How is this different from an arithmetic sequence?

A geometric sequence multiplies by a fixed amount each step; an arithmetic sequence adds a fixed amount. Multiplying versus adding is the difference.

How do I add up a geometric sequence?

Use the geometric series sum formula, which the sum of series calculator does for you. If the ratio is between -1 and 1, even an infinite number of terms has a finite total.

References

A note on where this comes from. A sequence with a constant ratio between terms is a geometric progression, a classical result in mathematics, and its nth-term formula, along with the finite and infinite sum formulas, are standard, set out in references such as Wolfram MathWorld. For further reading, see Geometric progression.

1. Weisstein, Eric W. "Geometric Series", Wolfram MathWorld, on the nth term and the sum of a geometric sequence.
2. Decimal.js, an arbitrary-precision decimal library, used here so repeated multiplication of decimals stays exact.