Arithmetic Sequence Calculator

aₙ = a₁ + (n−1)d. Find any term of an arithmetic sequence.

Inputs

Result

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How to use this calculator

  • Enter first term and common difference.
  • Pick term number n.
  • Read aₙ + sum.

About this calculator

Arithmetic sequence: each term differs from the previous by a constant d. Formula aₙ = a₁ + (n−1)d. Sum of first n terms: Sₙ = (n/2)(a₁ + aₙ). Examples: even numbers (a₁=2, d=2), powers-of-2 indices (a₁=1, d=1), or stair-step seating arrangements. Common in counting problems and discrete math.

Frequently asked

Difference vs ratio?+
Arithmetic: constant difference (d). Geometric: constant ratio (r). Different families.
Zero d?+
All terms equal a₁. Sum = n × a₁.
Negative d?+
Decreasing sequence. Eventually goes negative if n large enough.
Real-world?+
Salary raises (fixed dollar amt), stadium seating, library shelving, simple savings (fixed deposit).
Difference from sum-of-arithmetic-series?+
This calc finds individual term aₙ + bonus sum. Sum-of-series calc focuses on cumulative.

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Sₙ = n × (a₁ + aₙ) / 2 = n/2 × (2a₁ + (n−1)d).
Sum of Geometric Series
Sₙ = a₁(1 − rⁿ) / (1 − r) for r ≠ 1. Infinite: S∞ = a₁/(1−r) if |r|<1.
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F(n) = F(n−1) + F(n−2). 0, 1, 1, 2, 3, 5, 8, 13, 21, ...
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H = n / Σ(1/xᵢ). Used for averaging rates and ratios.
Weighted Average Calculator
WA = Σ(value × weight) / Σ(weight). Generalizes the arithmetic mean.

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