Sequence Calculator
Calculate sequence instantly with our math tool. Shows detailed work, formulas used, and multiple solution methods. Enter your values for instant results.
Reviewed for accuracy by Manoj Kumar, Mathematics Educator
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Formula: Arithmetic: aₙ = a₁ + (n-1)d, Sₙ = n(a₁+aₙ)/2 | Geometric: aₙ = a₁ × rⁿ⁻¹, Sₙ = a₁(1-rⁿ)/(1-r)
Worked example — a₁₀ = 48, Sum = 255
Arithmetic: aₙ = a₁ + (n-1)d, Sₙ = n(a₁+aₙ)/2 | Geometric: aₙ = a₁ × rⁿ⁻¹, Sₙ = a₁(1-rⁿ)/(1-r)
Arithmetic sequences add a constant difference. Geometric sequences multiply by a constant ratio.
Problem:a₁=3, d=5, n=10
Solution:a₁₀ = 3+9×5 = 48, Sum = 10×(3+48)/2 = 255
Result:a₁₀ = 48, Sum = 255
Arithmetic adds a constant (2,4,6,8...). Geometric multiplies by a constant (2,4,8,16...). Geometric sequences grow much faster.
Use the formula aₙ = a₁ + (n - 1)d, where a₁ is the first term, d is the common difference, and n is the term number. For example, in the sequence 5, 8, 11, 14... with d=3, the 20th term is 5 + (20-1)×3 = 5 + 57 = 62, without needing to list all 20 terms.
A geometric sequence with common ratio |r| < 1 converges to zero (terms shrink toward 0). With |r| > 1, the sequence diverges (terms grow without bound). With r = 1, all terms equal a₁. With r = -1, the sequence alternates between a₁ and -a₁. The infinite geometric series sum exists only when |r| < 1 and equals a₁ / (1 - r).
Arithmetic sequences appear in depreciation (fixed amount per year), savings with fixed contributions, and equally-spaced measurements. Geometric sequences model population growth, compound interest, radioactive decay, and virus spread. The key distinction is whether change is additive (arithmetic) or multiplicative (geometric).
Reviewed for accuracy by Manoj Kumar, Mathematics Educator · Editorial policy
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