Arithmetic and geometric sequence calculator
Give the first term and the step, or type a few terms of the sequence: we find the n-th term and the sum.
Result
21Term at position n (aₙ)
- Sum of the first n terms
- 120
- Common difference (what you add)
- 2
- First terms
- 3 ; 5 ; 7 ; 9 ; 11 ; 13
Inputs
- Type of sequence
- Arithmetic (+)
- I know
- a₁ and the step
- First term a₁
- 3
- Step
- 2
- Position n
- 10
How it is calculated
Arithmetic: aₙ = a₁ + (n − 1) × d, sum = n × (a₁ + aₙ) ÷ 2.
Result
Term at position n (aₙ)
- Sum of the first n terms
- 120
- Common difference (what you add)
- 2
- First terms
- 3 ; 5 ; 7 ; 9 ; 11 ; 13
How it is calculated
- Arithmetic: aₙ = a₁ + (n − 1) × d, sum = n × (a₁ + aₙ) ÷ 2.
- Geometric: aₙ = a₁ × rⁿ⁻¹, sum = a₁ × (1 − rⁿ) ÷ (1 − r) (or n × a₁ if r = 1).
- From terms: the step is the difference (or quotient) between neighbours; we check it is the same everywhere.
Example
Sequence 3, 5, 7… (step 2): a₁₀ = 3 + 9 × 2 = 21, and the first 10 terms add up to 120.
Good to know
- Several rules can give the same first terms: the result assumes an arithmetic or geometric sequence.
- The position is limited to 1,000,000; values that are too large are reported.
Frequently asked questions
What is this tool used for?
Give the first term and the step, or type a few terms of the sequence: we find the n-th term and the sum.
How is the result calculated?
Arithmetic: aₙ = a₁ + (n − 1) × d, sum = n × (a₁ + aₙ) ÷ 2.
What should I check before using the result?
Several rules can give the same first terms: the result assumes an arithmetic or geometric sequence.