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🔢 Sequence & Series Calculator

Arithmetic

Geometric

Fibonacci-style

Infinite Series

Formulas

Arithmetic Sequence

aₙ = a₁ + (n−1)d | Sₙ = n/2 · (2a₁ + (n−1)d)

First term (a₁)

Common difference (d)

Number of terms (n)

Geometric Sequence

aₙ = a₁ · rⁿ⁻¹ | Sₙ = a₁(1−rⁿ)/(1−r)

First term (a₁)

Common ratio (r)

Number of terms (n)

Fibonacci-Style Sequence

Each term = sum of the two before it: aₙ = aₙ₋₁ + aₙ₋₂

First term (a₁)

Second term (a₂)

Number of terms (n)

Infinite Geometric Series

S∞ = a₁/(1−r) when |r| < 1

First term (a₁)

Common ratio (r)

p-Series Test

Σ 1/nᵖ — converges if p > 1, diverges if p ≤ 1

Exponent (p)

Formula Reference Sheet

Arithmetic Sequence

          aₙ = a₁ + (n−1)d

          Sₙ = n/2 · (a₁ + aₙ)

          Sₙ = n/2 · (2a₁ + (n−1)d)

          d = aₙ − aₙ₋₁

Geometric Sequence

          aₙ = a₁ · rⁿ⁻¹

          Sₙ = a₁(1−rⁿ)/(1−r), r≠1

          S∞ = a₁/(1−r), |r|<1

          r = aₙ / aₙ₋₁

Convergence Tests

          Geometric: converges if |r|<1

          p-series Σ1/nᵖ: converges if p>1

          Harmonic Σ1/n: diverges (p=1)

          Ratio test: L=|aₙ₊₁/aₙ|; L<1→conv

Special Sums

          Σk = n(n+1)/2

          Σk² = n(n+1)(2n+1)/6

          Σk³ = [n(n+1)/2]²

          Σrᵏ (k=0→n) = (1−rⁿ⁺¹)/(1−r)