Arithmetic Sequence Calculator

An arithmetic sequence is a list of numbers where every consecutive pair differs by the same amount. That amount is the common difference. 3, 7, 11, 15, 19 has a common difference of 4. 100, 95, 90, 85 has a common difference of -5.

Two formulas cover everything:

nth term: aₙ = a₁ + (n − 1) × d

Sum of first n terms (Sₙ): Sₙ = n/2 × (a₁ + aₙ) (if you already know aₙ) Sₙ = n/2 × (2a₁ + (n − 1)d) (using only a₁ and d)

Both forms are identical — use whichever has fewer steps given what you know.

The Gauss trick: Carl Friedrich Gauss reportedly summed 1 + 2 + … + 100 in seconds as a schoolchild by pairing terms from each end: 1+100 = 101, 2+99 = 101, … 50 pairs of 101 = 5,050. That is exactly the sum formula: S = 100/2 × (1 + 100) = 5,050. The formula is just the Gauss trick written generally.

Worked example: A salary starts at $42,000 and increases by $2,500 per year. What is the salary in year 15, and what is the total paid over 15 years?

• a₁ = 42,000, d = 2,500, n = 15
• a₁₅ = 42,000 + (15 − 1) × 2,500 = 42,000 + 35,000 = $77,000
• S₁₅ = 15/2 × (42,000 + 77,000) = 7.5 × 119,000 = $892,500

When the sequence decreases: If d is negative, every term is smaller than the last. A vehicle purchased for $28,000 that loses $3,200/year in straight-line depreciation: a₁ = 28,000, d = -3,200. After 7 years: a₇ = 28,000 + 6 × (-3,200) = $8,800 book value.

Arithmetic vs geometric: Arithmetic sequences add a constant each step — linear growth on a graph. Geometric sequences multiply by a constant — exponential growth. Simple interest is arithmetic; compound interest is geometric. Over enough time the gap between them becomes enormous.

Reverse problems: Given aₙ but not n or d, rearrange:

• Find d if you know a₁, aₙ, and n: d = (aₙ − a₁) / (n − 1)
• Find n if you know a₁, d, and aₙ: n = (aₙ − a₁)/d + 1

Real uses that are actually arithmetic: Uniform acceleration in physics (velocity after t seconds at constant acceleration). Seat count in a wedge-shaped theater (each row adds a fixed number of seats). Tiling a staircase (row k needs k tiles; total for n rows = Sₙ). Any evenly-spaced measurement series.