Arithmetic Sequence Calculator – nth Term & Sum

Give the calculator a first term and a common difference and it produces any term without recursion. For 5, 9, 13, 17, … we have a₁ = 5 and d = 4, so the 50th term is a₅₀ = 5 + 49×4 = 201. The partial sum uses Gauss pairing — first plus last times count over two: S₅₀ = 50×(5+201)/2 = 50×103 = 5,150. The equivalent form Sₙ = n[2a₁+(n−1)d]/2 is shown too, useful when aₙ has not been computed yet. The tool accepts any two known terms instead: enter a₃ = 13 and a₇ = 29 and it derives d = 4, a₁ = 5, then rebuilds the sequence. Term index, value table and running sums are all printed, making it easy to lift the method straight into homework or a spreadsheet.

1. 1

   Enter a₁ and d

   Or supply any two terms — the calculator back-solves both

2. 2

   Choose n

   Pick the term index or the number of terms to sum

3. 3

   Apply the nth-term formula

   aₙ = a₁ + (n−1)d is evaluated with the substitution shown

4. 4

   Sum with Gauss pairing

   Sₙ = n(a₁+aₙ)/2 — both sum forms are displayed

• 1

  aₙ = a₁ + (n−1)d — the n−1 is where most answers go wrong

• 2

  Sₙ = n(a₁+aₙ)/2 works because paired terms always total a₁+aₙ

• 3

  Graphing terms against n gives a straight line with slope d

• Using n as the multiplier instead of n−1 in the nth-term formula

• Deriving d from two terms but forgetting the index gap, e.g. d = (a₇−a₃)/4 not /7

• Mixing arithmetic-sum formulas into geometric problems

• Plugging n = 0 and expecting the first term — sequences start at n = 1