Arithmetic and Geometric Sequence Calculator
Arithmetic sequences add a fixed difference: 3, 7, 11, … has 10th term 39.
39
- Sum of the first 10 terms
- 210
- First terms
- 3, 7, 11, 15, 19, 23, …
Step-by-step solution
- aₙ = a₁ + (n − 1)d = 3 + 9 × 4 = 39
- Sₙ = n(a₁ + aₙ)/2 = 10 × (3 + 39) ÷ 2 = 210
Arithmetic and Geometric Sequence Formula
aₙ = a₁ + (n − 1)d; aₙ = a₁rⁿ⁻¹
Sₙ = n(a₁ + aₙ)/2; Sₙ = a₁(1 − rⁿ)/(1 − r)
Worked Example
The arithmetic sequence 3, 7, 11, …:
- aₙ = a₁ + (n − 1)d = 3 + 9 × 4 = 39
- Sₙ = n(a₁ + aₙ)/2 = 10 × (3 + 39) ÷ 2 = 210
First terms: 3, 7, 11, 15, 19, 23, …
Test-day tip
Geometric sequences are exponential functions sampled at whole numbers; a growth question may be easier as aₙ = a₁rⁿ⁻¹.
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Matrix Calculator
det [[a, b], [c, d]] = ad − bc
A⁻¹ = (1/det) adj(A); no inverse when det = 0
Example: The matrix [[4, 7], [2, 6]]: Determinant = 10.
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Permutation and Combination Calculator
nPr = n!/(n − r)!, nCr = n!/(r!(n − r)!)
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Example: Choosing 3 items from 10: Combinations nCr = 120.
Frequently asked questions
How do I tell arithmetic and geometric sequences apart?
Subtract consecutive terms: a constant difference is arithmetic. Divide them: a constant ratio is geometric.
When does an infinite geometric series have a sum?
Only when |r| < 1; the sum is a₁ ÷ (1 − r).
Sources
Last reviewed: . Factors are stored exactly and computed with 50-digit decimal arithmetic. See how we verify conversion factors.
Cite this page
APA
Scientific Converters. (2026, October 15). Arithmetic and Geometric Sequence Calculator. Retrieved October 25, 2026, from https://scientificconverters.com/calculators/sequence-calculator/
IEEE
Scientific Converters, "Arithmetic and Geometric Sequence Calculator," scientificconverters.com. Accessed: Oct. 25, 2026. [Online]. Available: https://scientificconverters.com/calculators/sequence-calculator/