Number Sequence Calculator
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The 10th term is 39 and the sum of the first 10 terms is 210
Differences between neighboring terms: 4, 4, 4. They are all equal, so the sequence is arithmetic with d = 4. a₁₀ = a₁ + (n − 1)d = 3 + (10 − 1) × 4 = 39 S₁₀ = n ÷ 2 × (2a₁ + (n − 1)d) = 10 ÷ 2 × (2 × 3 + 9 × 4) = 210
| n | Term | Running sum |
|---|---|---|
| 1 | 3 | 3 |
| 2 | 7 | 10 |
| 3 | 11 | 21 |
| 4 | 15 | 36 |
| 5 | 19 | 55 |
| 6 | 23 | 78 |
| 7 | 27 | 105 |
| 8 | 31 | 136 |
| 9 | 35 | 171 |
| 10 | 39 | 210 |
How to use the number sequence calculator
- Choose the type of sequence. If you don't know it, pick Find the pattern and type the terms you have, such as
3, 7, 11, 15. - Otherwise enter the first term and the common difference (arithmetic), the common ratio (geometric) or the first two terms (Fibonacci-type).
- Enter n, the position of the term you want. You get that term, the sum of the first n terms, the nth-term formula and a table of the first terms with running totals.
Terms can be negative, decimals or fractions, and every result is exact. n can be up to 1 billion for arithmetic and quadratic sequences, 1,000 for geometric and 10,000 for Fibonacci-type, whose terms grow to thousands of digits. The table lists up to 100 terms.
Sequence formulas
Geometric: aₙ = a₁ × rn−1 · Sₙ = a₁(1 − rn) ÷ (1 − r) · S∞ = a₁ ÷ (1 − r) when |r| < 1
Fibonacci-type: aₙ = aₙ₋₁ + aₙ₋₂ · Sₙ = aₙ₊₂ − a₂
Quadratic: aₙ = a₁ + (n − 1)d₁ + (n − 1)(n − 2) ÷ 2 × d₂
Here aₙ is the term in position n, a₁ the first term, Sₙ the sum of the first n terms, d the common difference and r the common ratio. In a quadratic sequence d₁ is the first difference (a₂ − a₁) and d₂ the constant second difference. The Fibonacci sum rule works because adding the terms one by one builds the term two places further on, minus the second term.
Worked examples
Arithmetic: 3, 7, 11, 15, …
The difference is 4, so a₁₀ = 3 + 9 × 4 = 39 and S₁₀ = 10 ÷ 2 × (2 × 3 + 9 × 4) = 210. The formula simplifies to aₙ = 4n − 1.
Geometric: 3, 6, 12, … and 8, 4, 2, …
With r = 2, a₁₀ = 3 × 2⁹ = 1,536 and S₁₀ = 3 × (1 − 2¹⁰) ÷ (1 − 2) = 3,069. With r = 1/2 the terms shrink: the first 10 terms of 8, 4, 2, … add up to 15.984375, and the sum to infinity is 16.
Fibonacci: 1, 1, 2, 3, 5, …
a₁₀ = 55 and the first 10 terms add up to a₁₂ − a₂ = 144 − 1 = 143. The 100th term is 354,224,848,179,261,915,075.
Finding a pattern: 1, 4, 9, 16
The differences 3, 5, 7 are not constant, and neither are the ratios, but the second differences are all 2, so the sequence is quadratic: aₙ = n². The 20th term is 400 and the first 20 terms add up to 2,870.
Famous sequences: the first 12 terms
| n | Fibonacci | Lucas | Squares | Triangular | Powers of 2 |
|---|---|---|---|---|---|
| 1 | 1 | 2 | 1 | 1 | 1 |
| 2 | 1 | 1 | 4 | 3 | 2 |
| 3 | 2 | 3 | 9 | 6 | 4 |
| 4 | 3 | 4 | 16 | 10 | 8 |
| 5 | 5 | 7 | 25 | 15 | 16 |
| 6 | 8 | 11 | 36 | 21 | 32 |
| 7 | 13 | 18 | 49 | 28 | 64 |
| 8 | 21 | 29 | 64 | 36 | 128 |
| 9 | 34 | 47 | 81 | 45 | 256 |
| 10 | 55 | 76 | 100 | 55 | 512 |
| 11 | 89 | 123 | 121 | 66 | 1,024 |
| 12 | 144 | 199 | 144 | 78 | 2,048 |
Tips and common mistakes
- n counts from 1. The 10th term is 9 steps after the first, which is where the (n − 1) in the formulas comes from.
- A short list can fit several rules. 1, 2, 4 could be doubling (next 8) or a quadratic pattern (next 7). Give four or more terms to be sure.
- Geometric sums need r ≠ 1. When r = 1 every term is the same and Sₙ = n × a₁; the calculator handles this case for you.
- Growth is fast. Doubling 1 for 64 steps passes 18 quintillion, so exact big-number arithmetic matters. For powers on their own, use the exponent calculator; for numbers with hundreds of digits, the big number calculator.
Frequently asked questions
What is the formula for the nth term of an arithmetic sequence?
aₙ = a₁ + (n − 1)d, where a₁ is the first term and d is the common difference. For 3, 7, 11, 15, … the difference is 4, so the 10th term is 3 + 9 × 4 = 39, and in general aₙ = 4n − 1.
How do you add up an arithmetic sequence?
Use Sₙ = n ÷ 2 × (first term + last term). The whole numbers from 1 to 100 add up to 100 ÷ 2 × (1 + 100) = 5,050, and the first 10 terms of 3, 7, 11, … add up to 10 ÷ 2 × (3 + 39) = 210.
How do you find the sum of a geometric sequence?
Sₙ = a₁ × (1 − rⁿ) ÷ (1 − r) for any ratio r other than 1. For 3, 6, 12, … the first 10 terms add up to 3 × (1 − 2¹⁰) ÷ (1 − 2) = 3,069. When r is between −1 and 1 the terms shrink and the sum to infinity is a₁ ÷ (1 − r): 8 + 4 + 2 + … = 16.
What is the 100th Fibonacci number?
Counting F₁ = F₂ = 1, the 100th Fibonacci number is 354,224,848,179,261,915,075. It has 21 digits, which is why this calculator works with exact whole-number arithmetic instead of ordinary floating point.
How does the calculator find the pattern?
It tests the terms you enter for a constant difference (arithmetic), a constant ratio (geometric), each term being the sum of the two before (Fibonacci-type) and a constant second difference (quadratic, such as 1, 4, 9, 16), in that order. Several rules can fit a short list, so enter four or more terms when you can.