Arithmetic Sequence Calculator
The nth term and the sum of an arithmetic sequence.
Result
nth term
29.00
Sum of n terms
155.00
How it works
An arithmetic sequence adds the same amount at every step: start at a₁ and add the common difference d each time. The nth term needs no listing — jump straight there with aₙ = a₁ + (n−1)·d. Starting at 5 with d = 3, the 20th term is 5 + 19×3 = 62. The sum has an elegant shortcut, famously attributed to young Gauss adding 1 to 100: pair the first and last terms — each pair totals the same — so Sₙ = n·(a₁ + aₙ)/2. For 1 to 100: 100×101/2 = 5,050, in one line instead of ninety-nine additions. Arithmetic growth is everywhere something increases by a fixed amount per period: a savings plan adding the same deposit monthly, seat rows growing by a constant count, staircase heights. Contrast with geometric sequences, which multiply instead of add.
Advertisement
Frequently asked questions
What is the common difference?
The fixed amount added between consecutive terms — positive, negative or zero.
How is the sum found?
Multiply the number of terms by the average of the first and last term.
What is the story behind the sum formula?
Legend says schoolboy Gauss summed 1 to 100 in seconds by pairing ends: 1+100, 2+99… each pair makes 101, and there are 50 pairs — 5,050. The formula Sₙ = n(a₁+aₙ)/2 generalises exactly that trick.
Advertisement