SeqLab

Arithmetic Sequence Calculator

aₙ = a₁ + (n−1)d, Sₙ = (n/2)(2a₁ + (n−1)d).

First term a₁

Common difference d

Term index n

aₙ (nth term)

aₙ = a₁ + (n−1)d

Sₙ (sum of n terms): 155

What you entered

First term a₁: 2Common difference d: 3Term index n: 10

1
nth term: a₁ + (n−1)d
2 + (10−1) × 3= 29
2
Sum of first n terms
(10/2) × (2×2 + (10−1)×3)= 155

Result

aₙ (nth term): 29

Starting at 2 with common difference 3, term 10 is 29, and the first 10 terms sum to 155.

Arithmetic sequences: nth term and sum

Each term adds a constant difference to the one before. The nth term is a1 plus (n-1) times d, and the sum is n times the average of the first and last terms.

First term a1	Common difference d	Terms n	nth term	Sum of n terms
2	3	10	29	155
5	5	20	100	1050
1	2	100	199	10000
10	-2	8	-4	24
0	7	15	98	735

The third row is the classic result that the first 100 odd numbers sum to exactly 10,000 - a perfect square, which is true for any count of consecutive odd numbers starting at 1. Row four runs backwards with a negative difference, so the tenth term is below zero while the sum stays positive. The sum formula is n times the average of the first and last terms, which is why it is so much faster than adding the terms one by one: Gauss reportedly used it as a schoolboy to add 1 to 100 in seconds.

A constant step, every time

In an arithmetic sequence, each term differs from the last by the same fixed amount (the common difference d) — 2, 5, 8, 11… adds 3 every step, so the nth term is just the start plus (n−1) steps of size d.

Where it shows up

Simple interest, evenly-spaced seating rows, and any 'steady increase per period' scenario — like a savings plan that adds the same fixed amount every month — follow an arithmetic pattern.

Frequently asked questions

A theater has 20 seats in row 1 and adds 3 seats per row. How many seats are in row 15?

This is an arithmetic sequence with a₁ = 20 and d = 3. The 15th term: a₁₅ = 20 + (15 − 1) × 3 = 20 + 42 = 62 seats in row 15.

How is an arithmetic sequence different from a geometric one?

Arithmetic adds the same amount each step (2, 5, 8, 11 — adding 3). Geometric multiplies by the same amount each step (2, 6, 18, 54 — multiplying by 3). Arithmetic grows linearly; geometric grows exponentially. Use the geometric sequence calculator for multiplicative patterns.

How do I find the sum of the first 100 positive integers?

This is the arithmetic series 1 + 2 + 3 + … + 100 with a₁ = 1, d = 1, n = 100. Gauss's formula: S = n(a₁ + aₙ)/2 = 100(1 + 100)/2 = 5,050. The calculator shows this step by step.

Can an arithmetic sequence have negative terms?

Yes — either the first term is negative, or the common difference is negative (a decreasing sequence). For example, 10, 7, 4, 1, −2, −5 is arithmetic with d = −3. The formulas work identically regardless of sign.