Number Sequence Calculator
Fractions & numbersFreeNo sign-up
Type the first few terms and the calculator finds the rule. 3, 7, 11, 15 is arithmetic (add 4 each time), so the nth term is an = 4n − 1 and the 10th term is 39.
Type the problems from this lesson and check each step.
| Sequence | |
| Terms given | 3, 7, 11, 15 |
| Pattern | |
| Type | Arithmetic |
| Rule | aₙ = 4n − 1 |
| Next five terms | 19, 23, 27, 31, 35 |
| Answers | |
| Term 10 | 39 |
| Sum of the first 10 terms | 210 |
Show steps
- The differences are all 4, so the sequence is arithmetic with d = 4.
- aₙ = a₁ + (n − 1)d = 3 + (n − 1) × 4.
How to use the calculator
- Type at least three terms in order (four or five for quadratic and cubic patterns).
- Choose which term number n you want.
- Press Calculate to see the type of sequence, its rule, the next five terms, term n and the sum of the first n terms.
Types of sequences
| Type | Example | How to spot it | nth term |
|---|---|---|---|
| Arithmetic | 3, 7, 11, 15 | Same difference | a₁ + (n − 1)d |
| Geometric | 2, 6, 18, 54 | Same ratio | a₁ × rⁿ⁻¹ |
| Quadratic | 2, 5, 10, 17 | Same second difference | an² + bn + c |
| Fibonacci-type | 1, 1, 2, 3, 5 | Add the two before | aₙ₋₁ + aₙ₋₂ |
Sum formulas
- Arithmetic: Sₙ = n ÷ 2 × (a₁ + aₙ). The sum 1 + 2 + … + 100 = 100 ÷ 2 × 101 = 5,050.
- Geometric (r ≠ 1): Sₙ = a₁(1 − rⁿ) ÷ (1 − r). 1 + 2 + 4 + … + 512 = 1 × (1 − 2¹⁰) ÷ (1 − 2) = 1,023.
A few terms can fit more than one rule. The calculator picks the simplest one that fits every term you entered.
Number sequence practice questions
Test yourself with 6 multiple-choice questions. Choose an answer and press Submit to see if you are right and why. Your score appears at the end.
Quiz complete
Frequently asked questions
How do you find the nth term of an arithmetic sequence?
Use aₙ = a₁ + (n − 1)d, where d is the common difference. For 3, 7, 11, …, aₙ = 3 + 4(n − 1) = 4n − 1.
How do you know if a sequence is geometric?
Divide each term by the one before it. If you always get the same number, the sequence is geometric and that number is the common ratio.
What if the differences are not constant?
Find the differences of the differences. If the second differences are constant the rule is quadratic, and the coefficient of n² is half the second difference.
Can a sequence have more than one rule?
Yes. Any finite list can be continued in many ways. The calculator gives the simplest common rule that fits all the terms you entered.
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