Number Sequence Calculator
Analyze number sequences, find patterns, and calculate nth terms.
Sequence Type
Sequence Type
Each term increases by 2
Term 5
nth term
Sum
First 10 terms
Sequence Terms
Formula & Pattern
How it works
A number sequence calculator finds terms and sums of patterned sequences. Arithmetic sequences add a constant difference each step; geometric sequences multiply by a constant ratio. Each has a formula for the nth term and the running total.
Arithmetic & geometric
Arithmetic: aₙ = a₁ + (n−1)d Geometric: aₙ = a₁ · r^(n−1)
- a₁
- first term
- d
- common difference (arithmetic)
- r
- common ratio (geometric)
- n
- term position
Worked example
- Arithmetic: start 3, difference 5
- Find the 6th term
- a₆ = 3 + (6 − 1) × 5
- a₆ = 3 + 25
The 6th term is 28 (sequence: 3, 8, 13, 18, 23, 28).
Good to know
- Arithmetic sequences grow in a straight line; geometric ones grow exponentially.
- The sum of an arithmetic series is n × (first + last) ÷ 2.
- A geometric series with |r| < 1 converges to a finite sum even with infinite terms.
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Frequently Asked Questions
What is an arithmetic sequence?
A sequence with a constant difference between consecutive terms, like 3, 7, 11, 15 (difference 4). The nth term is aₙ = a₁ + (n−1)d, and the sum of the first n terms is n(a₁ + aₙ) ÷ 2.
What is a geometric sequence?
A sequence where each term is the previous one multiplied by a constant ratio, like 2, 6, 18, 54 (ratio 3). The nth term is aₙ = a₁ × rⁿ⁻¹. Geometric growth appears in compound interest, population models, and radioactive decay.
How do I find the pattern in a sequence?
Check differences between terms first — constant differences mean arithmetic. Then check ratios — constant ratios mean geometric. If neither, try second differences (constant for quadratic patterns) or look for special sequences like Fibonacci, squares, cubes, or primes.
What is the Fibonacci sequence?
Each term is the sum of the two before it: 0, 1, 1, 2, 3, 5, 8, 13, … The ratio of consecutive terms approaches the golden ratio (≈1.618), and the pattern appears in nature — branching, spirals, and seed arrangements.
How do I sum a sequence?
Arithmetic series: Sₙ = n(a₁ + aₙ)/2. Geometric series: Sₙ = a₁(1 − rⁿ)/(1 − r) for r ≠ 1; if |r| < 1, the infinite sum converges to a₁/(1 − r). Identify the sequence type first, since the formulas aren't interchangeable.