Arithmetic and geometric sequences
The nth term and the sum of a sequence, with the first terms listed.
How it works
Arithmetic: uₙ = u₁ + (n − 1) d and Sₙ = n (u₁ + uₙ) / 2. Geometric: uₙ = u₁ × rⁿ⁻¹ and Sₙ = u₁ (1 − rⁿ) / (1 − r).
Results are for guidance only and do not replace professional advice.
Worked example
With the default values:
- First term u₁3
- Common difference or ratio2
- Term number n10
- Sequence typeGeometric (× r)
you get:
- Term uₙ (n = 10)1,536
- Sum of the first 10 terms3,069
- First terms3 ; 6 ; 12 ; 24 ; 48 ; 96 ; 192 ; 384 ; 768 ; 1,536
- Formulauₙ = 3 × 2ⁿ⁻¹
Frequently asked questions
How does the “Arithmetic and geometric sequences” tool work?
Arithmetic: uₙ = u₁ + (n − 1) d and Sₙ = n (u₁ + uₙ) / 2. Geometric: uₙ = u₁ × rⁿ⁻¹ and Sₙ = u₁ (1 − rⁿ) / (1 − r).
What do I need to enter?
Just fill in: first term u₁, common difference or ratio, term number n, sequence type. The result updates as you type.
Can you give an example?
First term u₁: 3; Common difference or ratio: 2; Term number n: 10; Sequence type: Geometric (× r) → Term uₙ (n = 10): 1,536; Sum of the first 10 terms: 3,069.
Is this calculator free, and is my data private?
Yes. HOD Calc is completely free with no sign-up. The calculation runs in your browser: nothing you type is sent or stored.