Math

Arithmetic Sequence Calculator (aₙ = a₁ + (n−1)d)

Enter the first term a₁ and common difference d to compute the nth term and partial sums of an arithmetic sequence. Includes a term table and bar chart.

Tips

  • A positive d produces an increasing sequence (e.g. 1, 3, 5, 7, …), a negative d produces a decreasing one, and d = 0 gives a constant sequence.
  • The nth term is aₙ = a₁ + (n − 1)d and the sum of the first n terms is Sₙ = n(a₁ + aₙ) / 2.
  • Type an integer into the "n value" field to instantly compute the nth term and cumulative sum — handy for checking homework answers.
  • Each bar in the chart corresponds to one term. Equal bar height differences confirm you have a true arithmetic sequence.

FAQ

d is the constant difference between consecutive terms (next term − previous term). For example, in 2, 5, 8, 11, … the common difference is d = 3.

Sₙ = n(a₁ + aₙ) / 2, which equals "average of first and last term × number of terms". It can also be written as Sₙ = n(2a₁ + (n−1)d) / 2.

In an arithmetic sequence the difference between consecutive terms is constant. In a geometric sequence the ratio between consecutive terms is constant. For example, 2, 4, 6, 8 is arithmetic (d = 2) while 2, 4, 8, 16 is geometric (r = 2).
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Side Note — Gauss and the legend of 1 to 100

The partial-sum formula has a famous origin story. When the mathematician Carl Friedrich Gauss was a schoolboy, his teacher asked the class to add all integers from 1 to 100. While other students laboriously added one by one, Gauss noticed that 1 + 100 = 101, and there are 50 such pairs, giving 5 050 instantly.

That insight — "the first and last terms sum to the same value" — is exactly the formula Sₙ = n(a₁ + aₙ) / 2. Gauss went on to be called the "Prince of Mathematics" and contributed to the normal distribution, prime number theorem, and electromagnetism, but his sharpness was clearly evident from childhood.

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