Geometric Sequence Calculator
Enter the first term and the common ratio to get any term of a geometric sequence, the sum of the first n terms, and — when the ratio is between −1 and 1 — the sum of the whole infinite series. Or give two terms you know and the calculator works back to a₁ and r. Answers are exact fractions, and each formula is shown with your numbers substituted. Free, no sign-up.
Multiply by this each time. Negative makes the signs alternate.
A whole number from 1 to 2000.
Results
1536
Term a₁₀
3069
Sum S₁₀ of the first 10 terms
No finite sum
Sum of all terms, S∞
2
Common ratio
3
First term
|r| is greater than 1, so the terms grow without bound.
General formula
aₙ = 3 · 2^(n − 1)
Working
a₁₀ = a₁ · r^(n − 1) = 3 · 2^(10 − 1) = 1536
S₁₀ = a₁(1 − r^n) ÷ (1 − r) = 3 × (1 − 2^10) ÷ (1 − 2) = 3069
The infinite series converges only when |r| < 1. With |r| of 1 or more the terms do not shrink, so the sum has no finite value.
The sequence
3, 6, 12, 24, 48, 96, 192, 384, …, 1536
How to Use the Geometric Sequence Calculator
- 1. Choose what you know. First term and common ratio is the usual case. Two terms of the sequence handles questions like 'the 2nd term is 6 and the 5th is 162'. Find where a value sits tells you which term equals a number.
- 2. Enter the values. Whole numbers, decimals and fractions such as 1/2 all work. The ratio can be negative or a fraction. Term numbers run from 1 to 2000.
- 3. Read the cards. You get the term aₙ, the sum Sₙ, the sum to infinity — or No finite sum when |r| ≥ 1 — the ratio and the first term. Values with more than 40 digits are shown in scientific notation.
- 4. Follow the working. The formulas are written out with your numbers, followed by the first few terms.
The Formulas
A geometric sequence multiplies by the same common ratio r to get from each term to the next: a₁, a₁r, a₁r², …. So the n-th term and the sum of the first n terms are:
aₙ = a₁ · r^(n − 1) Sₙ = a₁(1 − rⁿ) ÷ (1 − r) (r ≠ 1) Sₙ = n · a₁ (r = 1) S∞ = a₁ ÷ (1 − r) (only when |r| < 1)
The sum formula comes from subtracting r·Sₙ from Sₙ: nearly every term cancels, leaving Sₙ(1 − r) = a₁(1 − rⁿ). When |r| < 1, rⁿ shrinks toward 0 as n grows, so the partial sums settle at a₁ ÷ (1 − r). When |r| ≥ 1 they do not settle, and the infinite series has no finite sum.
Worked Examples
- Term and sum. For 3, 6, 12, … a₁ = 3 and r = 2. The 10th term is 3 × 2⁹ = 1536 and the sum of the first 10 is 3(1 − 2¹⁰) ÷ (1 − 2) = 3069.
- From two terms. If a₂ = 6 and a₅ = 162, then r³ = 162 ÷ 6 = 27, so r = 3. Then a₁ = 6 ÷ 3 = 2, and the sequence is 2, 6, 18, 54, 162, …. When the two terms are an even number of steps apart, r² = 4 for example, there are two real answers (2 and −2), and the calculator shows both.
- Infinite sum. For 8, −8/3, 8/9, … a₁ = 8 and r = −1/3, so S∞ = 8 ÷ (1 + 1/3) = 6. The same formula shows why 0.999… = 1: it is 0.9 + 0.09 + 0.009 + … with a₁ = 0.9 and r = 0.1, so S∞ = 0.9 ÷ 0.9 = 1.
- The chessboard. One grain on the first square, doubling on each of the 64 squares: a₁ = 1, r = 2, n = 64, so S₆₄ = 2⁶⁴ − 1 = 18446744073709551615 grains.
| Ratio r | What the sequence does | Infinite sum |
|---|---|---|
| r > 1 | Grows without bound | None |
| r = 1 | Constant | None (unless a₁ = 0) |
| 0 < r < 1 | Shrinks toward 0 | a₁ ÷ (1 − r) |
| −1 < r < 0 | Alternates in sign and shrinks | a₁ ÷ (1 − r) |
| r = −1 | Alternates between a₁ and −a₁ | None |
| r < −1 | Alternates and grows | None |
Frequently Asked Questions
- What is the formula for the nth term of a geometric sequence?
- aₙ = a₁ · r^(n − 1), where a₁ is the first term and r is the common ratio. For 5, 10, 20, … it gives aₙ = 5 · 2^(n − 1), so the 8th term is 5 × 128 = 640.
- How do you find the common ratio?
- Divide any term by the one before it. In 4, 12, 36, 108 each division gives 3. If you only know two terms that are not neighbours, take the k-th root of their quotient, where k is the number of steps between them: with a₂ = 6 and a₅ = 162, r = ∛(162 ÷ 6) = 3.
- When does an infinite geometric series have a sum?
- Exactly when |r| < 1 — the ratio is between −1 and 1, not including them. Then S∞ = a₁ ÷ (1 − r). For 1 + 1/2 + 1/4 + … that is 1 ÷ (1/2) = 2. If |r| ≥ 1 the terms do not shrink and the sum has no finite value.
- What is the difference between arithmetic and geometric sequences?
- An arithmetic sequence adds a constant (2, 5, 8, 11); a geometric one multiplies by a constant (2, 6, 18, 54). Differences are constant in the first; ratios are constant in the second. The Number Sequence Solver tests both for you.
- Can the common ratio be negative or a fraction?
- Yes. A negative ratio makes the signs alternate (3, −6, 12, −24) and a fraction between −1 and 1 makes the terms shrink (81, 27, 9, 3, 1, 1/3). A ratio of 0 is allowed here, but every term after the first is then 0.
- What does the sum formula give when r = 1?
- The formula a₁(1 − rⁿ) ÷ (1 − r) would divide by zero, so use Sₙ = n · a₁ instead: every term equals a₁, and there are n of them. The calculator switches to this form automatically.
- Where do geometric sequences appear in real life?
- Compound interest is geometric: 1000 at 5% a year is worth 1000 · 1.05ⁿ after n years. Population growth, depreciation, radioactive decay and the bounce heights of a ball all follow the same pattern with a ratio above or below 1.