Arithmetic Sequence Calculator

How to Use This Calculator

1. Enter the first term a₁, common difference d, and n.
2. The nth term and sum Sₙ are computed instantly.
3. Use Find Common Difference tab to derive d from two known terms.

Formula

Example

Frequently Asked Questions

• What is an arithmetic sequence? ▼
  An arithmetic sequence (also called an arithmetic progression) is an ordered list of numbers in which each term after the first is obtained by adding a fixed constant called the common difference d. The general form is: a₁, a₁+d, a₁+2d, a₁+3d, …. Example: 3, 7, 11, 15, 19, … has a₁ = 3 and d = 4. To confirm it is arithmetic, check that consecutive differences are equal: 7−3 = 4, 11−7 = 4, 15−11 = 4. A common pitfall is confusing arithmetic sequences (constant difference) with geometric sequences (constant ratio). In an arithmetic sequence d can be negative — for example 20, 15, 10, 5, 0, −5, … is arithmetic with d = −5.
• What is the formula for the nth term? ▼
  The nth term of an arithmetic sequence is aₙ = a₁ + (n − 1) × d, where a₁ is the first term, d is the common difference, and n is the term number. Example: a₁ = 3, d = 4, find a₁₀. a₁₀ = 3 + (10 − 1) × 4 = 3 + 36 = 39. Another example: find the 25th term of 100, 93, 86, …. Here a₁ = 100, d = −7, so a₂₅ = 100 + 24 × (−7) = 100 − 168 = −68. A common mistake is using n instead of (n − 1) in the formula — that shifts every result by one d unit. Also note that if you know two terms aₘ and aₙ, you can find d = (aₙ − aₘ)/(n − m).
• What is the sum formula for arithmetic sequences? ▼
  The sum of the first n terms of an arithmetic sequence is Sₙ = n/2 × (2a₁ + (n − 1)d), which can also be written as Sₙ = n/2 × (a₁ + aₙ). The second form says: multiply the number of terms by the average of the first and last term. Example: a₁ = 3, d = 4, n = 10. First find a₁₀ = 3 + 9 × 4 = 39. Then S₁₀ = 10/2 × (3 + 39) = 5 × 42 = 210. This is Gauss's trick — pairing terms from opposite ends each gives the same sum. A common pitfall is off-by-one in counting: "the sum from 1 to 100" has 100 terms, not 99. Always verify by checking S₁ = a₁ and S₂ = a₁ + a₂.
• Does an arithmetic series converge? ▼
  No — an arithmetic series (the infinite sum a₁ + a₂ + a₃ + …) always diverges unless d = 0. When d ≠ 0, the terms grow without bound in magnitude, so the partial sums Sₙ tend to +∞ (if d > 0) or −∞ (if d < 0). Even with d = 0 (a constant sequence), the infinite sum diverges to ±∞ unless a₁ = 0. By contrast, a geometric series with |r| < 1 does converge to a finite sum. A common misconception is thinking that because the partial sum formula Sₙ exists, there must be a limiting value — but Sₙ grows as n², not approaching any fixed number. Arithmetic sequences are only summed over finite ranges in practice.
• What are real-world examples of arithmetic sequences? ▼
  Arithmetic sequences appear whenever a quantity increases or decreases by a fixed amount each period. Straight-line depreciation: a machine worth $10,000 loses $1,000 per year, giving values $10,000, $9,000, $8,000, …. Equal principal mortgage payments: if $200 of principal is repaid each month, the outstanding balance decreases arithmetically. Seating in a theater: row 1 has 20 seats, row 2 has 22, row 3 has 24, … — each row adds 2 seats. Stacking bricks: each layer has one fewer brick than the layer below. A common pitfall is applying arithmetic formulas to compound-interest growth, which is geometric (multiplied by a factor), not arithmetic (added a constant). Always check whether the constant is additive or multiplicative.

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