Arithmetic Sequence Calculator

Last updated: 2026-05-19

What Is an Arithmetic Sequence?

An arithmetic sequence is a list of numbers where each term is obtained by adding a fixed amount — the common difference d — to the previous term. If the first term is a₁, the sequence looks like:

$$a_1,\quad a_1 + d,\quad a_1 + 2d,\quad a_1 + 3d,\quad \ldots$$

The calculator finds three things in one step: the nth term (aₙ), the sum of the first n terms (Sₙ), and the average of those n terms.

The nth Term Formula

The nth term of an arithmetic sequence is:

$$a_n = a_1 + (n - 1) \cdot d$$

Why this works: each step from a₁ to aₙ adds one copy of d. Reaching position n takes n − 1 steps, so d is added exactly n − 1 times.

Worked example. The sequence 3, 7, 11, 15, … has a₁ = 3 and d = 4. The 20th term is:

$$a_{20} = 3 + (20 - 1) \times 4 = 3 + 76 = 79$$

The Sum Formula and the Gauss Pairing Trick

The sum of the first n terms is:

$$S_n = \frac{n \cdot (a_1 + a_n)}{2}$$

The formula follows from pairing terms. Writing the sum forwards and backwards, the first term pairs with the last, the second with the second-to-last, and so on; each pair has the same total. For the integers from 1 to 100, the pairs are:

$$1 + 100 = 101, \quad 2 + 99 = 101, \quad 3 + 98 = 101, \quad \ldots$$

There are 100/2 = 50 such pairs, each summing to 101, giving 50 × 101 = 5050. In general there are n/2 pairs of (a₁ + aₙ) each, hence Sₙ = n(a₁ + aₙ)/2. The pairing argument is commonly attributed to Carl Friedrich Gauss.

An equivalent form that avoids computing aₙ first:

$$S_n = \frac{n \cdot (2a_1 + (n-1)d)}{2}$$

The Average

The average of the first n terms equals the arithmetic mean of the first and last:

$$\bar{a} = \frac{S_n}{n} = \frac{a_1 + a_n}{2}$$

This is exactly the midpoint of the sequence — a consequence of the terms being evenly spaced.

Connection to Linear Functions

Arithmetic sequences are the discrete version of linear functions. Plotting the term index n on the x-axis and the term value aₙ on the y-axis places the points on the line:

$$y = a_1 + (x - 1) \cdot d$$

The common difference d plays the role of the slope, and a₁ is the y-intercept (shifted by one, since sequences start at n = 1 rather than n = 0). This connection makes arithmetic sequences the natural starting point for understanding linear relationships.

Special Cases

Condition	Effect
d = 0	Constant sequence; every term equals a₁; Sₙ = n·a₁
d < 0	Decreasing sequence; terms eventually become negative
d fractional	Perfectly valid — terms are equally spaced decimals
n = 1	aₙ = a₁; S₁ = a₁; average = a₁

Worked Example with Realistic Values

A variable-rise staircase has 12 steps. The first step is 16 cm tall and each step is 0.5 cm taller than the one before it (d = 0.5). The height of the 12th step and the total rise are found as follows:

• a₁ = 16, d = 0.5, n = 12
• a₁₂ = 16 + 11 × 0.5 = 21.5 cm
• S₁₂ = 12 × (16 + 21.5)/2 = 12 × 18.75 = 225 cm total rise
• Average step height = 18.75 cm

Frequently Asked Questions (FAQ)

How do I find the nth term of an arithmetic sequence?

Use the formula aₙ = a₁ + (n − 1)·d, where a₁ is the first term, d is the common difference, and n is the position. For example, in the sequence 2, 5, 8, 11, … (a₁ = 2, d = 3), the 10th term is 2 + (10 − 1)·3 = 2 + 27 = 29.

What's the formula for the sum of an arithmetic sequence?

The sum of the first n terms is Sₙ = n·(a₁ + aₙ)/2. This is the Gauss pairing trick: pair the first and last terms (a₁ + aₙ), the second and second-to-last, and so on — each pair has the same sum, and there are n/2 pairs. An equivalent form is Sₙ = n·(2a₁ + (n − 1)d)/2.

How is an arithmetic sequence different from a geometric sequence?

An arithmetic sequence adds a fixed amount (the common difference d) at each step, producing linear growth. A geometric sequence multiplies by a fixed ratio r at each step, producing exponential growth or decay. Example: 2, 5, 8, 11 is arithmetic (d = 3); 2, 6, 18, 54 is geometric (r = 3).

Can the common difference be negative or fractional?

Yes — d can be any real number. A negative d gives a decreasing sequence (e.g., 10, 7, 4, 1, −2, …). A fractional d is equally valid (e.g., d = 0.5 gives 1, 1.5, 2, 2.5, …). When d = 0 every term equals a₁ and the sum simplifies to Sₙ = n·a₁.