Calculate arithmetic and geometric sequences, find nth terms, and compute series sums with step-by-step solutions.
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aₙ = a₁ + (n−1)d; Sₙ = n/2 · [2a₁ + (n−1)d]; Partial sums S(m..n) = Sₙ − S₍ₘ₋₁₎.
aₙ = a₁ · r^(n−1); Sₙ = a₁ (1 − r^n)/(1 − r) for r≠1; if r=1 then Sₙ = n·a₁.
Infinite sum S∞ = a₁/(1 − r) when |r| < 1.
If second difference is constant, t(n) = a n² + b n + c. Fit using first three terms: a = (t3 − 2 t2 + t1)/2, b = (t2 − t1) − 3a, c = t1 − a − b.
Sum: Sₙ = a n(n+1)(2n+1)/6 + b n(n+1)/2 + c n.
Arithmetic, geometric, quadratic, or use detection.
Generate terms, nth term, sums, parameters, position from value, or infinite sum (geometric).
Supply a₁ and a₂ (to infer d or r), or first three terms for quadratic; enter indices as needed.
See authentic formulas and step-by-step derivations.
Arithmetic grows linearly with n; geometric grows or decays exponentially with ratio r.
Geometric S∞ converges only if |r|<1; otherwise the series diverges.
Constant second differences characterize quadratic sequences; sums are cubic in n.
Work with arithmetic, geometric, and quadratic sequences. Generate terms, compute nth values, sums, partial sums, detect patterns, and more.
Calculate arithmetic sequences, geometric series, quadratic patterns, and infinite sums with our comprehensive sequence and series calculator. Generate terms, find nth values, compute partial sums, detect patterns, and solve for parameters. Perfect for students, mathematicians, and professionals working with mathematical sequences, series convergence, and pattern analysis.
Support for arithmetic sequences (constant difference), geometric series (constant ratio), quadratic patterns (second differences), and automatic pattern detection.
Generate sequences, find nth terms, calculate sums of first n terms, partial sums, infinite geometric series, and solve for unknown parameters.
Automatically detect arithmetic, geometric, or quadratic patterns from a list of terms with detailed analysis and parameter identification.
Sequence calculator, series calculator, arithmetic sequence, geometric series, quadratic sequence, infinite series, pattern detection, mathematical sequences, nth term calculator, sum calculator, convergence analysis, sequence formulas.
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Important: This sequence and series calculator is designed for educational, mathematical analysis, and problem-solving purposes using established algebraic formulas for arithmetic sequences, geometric sequences, and quadratic patterns. While calculations employ standard mathematical algorithms and closed-form formulas with appropriate rounding for display purposes, the tool is intended for learning and verification rather than advanced mathematical research. For academic submissions, professional mathematical work, or research applications requiring high precision, verify results using multiple methods and consult mathematical references. The calculator handles standard sequence types but may not account for specialized mathematical contexts or advanced series convergence considerations that could be relevant for higher-level mathematical analysis.