Introduction
A geometric sequence is a list of numbers where you multiply by the same number each time. That number is called the common ratio, or r. For example, in 2, 6, 18, 54, each term is 3 times the one before it.
Type in what you know, and this Geometric Sequence Calculator finds:
- The nth term formula, like aₙ = a₁ · rⁿ⁻¹
- Any term you pick, such as the 5th or 20th term
- The next term in a list you already have
- The common ratio r or the first term a₁
- The sum of the first n terms (finite sum)
- The sum of all terms forever (infinite sum), when it converges
It also shows step-by-step work, a table of terms with running sums, a graph, and a number line. And it tells you if the series converges (adds up to one number) or diverges (grows without end).
You can enter whole numbers, decimals, or fractions like 1/2. Check one box to get exact fractions in your answer. If you are not sure where to start, tap a quick preset or an example to load a real problem in one click. There is also a checker where you can type your own answer and see if you got it right.
How to use our Geometric Sequence Calculator
Enter what you know about your geometric sequence, such as the first term, the common ratio, or a few terms, and the calculator finds the nth term, the next term, partial sums, and the infinite sum, with step-by-step work, a table, a graph, and a number line.
Quick Pick Presets: Click a preset like Growth, Decay Sum, or Convergent ∞ to fill the form with a ready-made sequence and see the answer right away.
Solve For: Pick what you want to find: the nth term formula, a specific term, the next term, the common ratio r, the first term a₁, a finite sum Sₙ, or an infinite sum S∞.
Known Terms: Type at least two terms of your sequence, split by commas, like 2, 10, 50. Fractions like 1/2 work too.
Direct Formula: If you already have a closed form, type it here, like 3*5^(n-1). The tool reads the first term and ratio from it.
a₁ (First Term): Type the first term of the sequence. You can use a decimal or a fraction, like -3/4.
r (Common Ratio): Type the number you multiply by to get the next term, like 3 or 1/2. Use a minus sign for an alternating sequence.
Find term at position n: Type the term number you want, from 1 to 1000. The calculator shows the value of that term.
Known Infinite Sum S∞: If you know the sum of all terms, type it here. This only works when |r| is less than 1.
Known Terms at Specific Positions: Type a position n and its value a(n). Click "Add Another Term" to enter a second pair so the tool can solve for r and a₁.
Known or Requested Partial Sums: Type how many terms n you want summed. Leave S(n) blank to compute it, or type a known sum to solve for missing values.
Find Common Ratio r from Two Terms: Type two term values and their index numbers. The calculator divides and takes a root to find r.
Math Symbol Pad: Click it to insert fractions, powers, roots, or the infinity sign into the last math box you clicked.
Output Options: Check the boxes to show the steps, the terms table, the graph, and the number line. Pick exact fractions for exact answers, or set rounding and decimal places for decimals.
Calculate and Clear All: Press Calculate to get your answer, or Clear All to empty every box and start over.
Check Your Own Answer: Open the answer checker, type your own answer, and press Check to see if it matches.
What Is a Geometric Sequence?
A geometric sequence is a list of numbers where you multiply by the same number each time to get the next term. That number is called the common ratio, written as r. The first term is called a₁.
Example: 2, 6, 18, 54, … Here you multiply by 3 every time, so r = 3 and a₁ = 2.
How to Find the Common Ratio
Divide any term by the term right before it:
r = a₂ ÷ a₁
In 2, 6, 18: r = 6 ÷ 2 = 3. If the ratio is the same for every pair, the list is geometric. If not, it is some other kind of sequence.
The nth Term Formula
To jump straight to any term without listing them all, use:
aₙ = a₁ × r(n−1)
Example: with a₁ = 2 and r = 3, the 5th term is 2 × 34 = 2 × 81 = 162.
Adding the Terms: Geometric Series
When you add the terms together, you get a geometric series.
- Sum of the first n terms: Sₙ = a₁ × (1 − rn) ÷ (1 − r)
- Special case r = 1: every term is the same, so Sₙ = n × a₁
- Sum of infinite terms (only if |r| < 1): S∞ = a₁ ÷ (1 − r)
When Does an Infinite Series Converge?
An infinite geometric series only adds up to a real number when the ratio is small, meaning |r| < 1. Then the terms shrink toward zero and the total settles on one value. We say the series converges.
If |r| is 1 or bigger, the terms stay big or grow, and the sum keeps getting larger forever. We say the series diverges, and there is no infinite sum.
Example: 6, 2, 2/3, … has r = 1/3, so S∞ = 6 ÷ (1 − 1/3) = 9.
What the Ratio Tells You
| Value of r | What the sequence does |
|---|---|
| r > 1 | Grows fast (exponential growth) |
| 0 < r < 1 | Shrinks toward zero (decay) |
| r = 1 | Stays the same forever |
| r < 0 | Flips between positive and negative (alternating) |
| r = 0 | Drops to zero after the first term |
Geometric vs. Arithmetic Sequences
In an arithmetic sequence you add the same number each time (2, 5, 8, 11). In a geometric sequence you multiply by the same number each time (2, 6, 18, 54). Multiplying makes geometric sequences change much faster.
Where Geometric Sequences Are Used
- Money: compound interest grows by the same percent each year.
- Science: half-life of a radioactive sample cuts in half each period (r = 1/2).
- Biology: cells or bacteria that double (r = 2).
- Computers: memory sizes and data that double at each step.
- Sound and bouncing balls: each bounce reaches a fixed fraction of the last height.
Quick Tips
- You only need two pieces of info, like a₁ and r, to build the whole sequence.
- If you know two terms at known spots, divide them and take a root to find r.
- Watch the signs: a negative r makes every other term negative.
- Fractions like 1/2 or 1/3 give exact answers, so keep them as fractions when you can.