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Generated on: July 20, 2026 at 09:33 PM

Arithmetic Sequence Calculator

Formula

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

Inputs

First term2
Common difference3
Term number10

Visualization

Term aⱼ 0

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Mathematics

Arithmetic Sequence Calculator

Find the nth term and sum of an arithmetic sequence. Enter the first term, common difference, and n to get aₙ, Sₙ, and the average.

Inputs

Sequence Parameters

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

Results

Enter a value to see results.

Details

Results

\begin{aligned} a_n &= a_1 + (n - 1)d \\ &= 2 + (10 - 1)(3) \\ &= ? \end{aligned}
\begin{aligned} S_n &= \dfrac{n(a_1 + a_n)}{2} \\ &= \dfrac{10(2 + ?)}{2} \\ &= ? \end{aligned}
\begin{aligned} \bar{a} &= \dfrac{S_n}{n} \\ &= \dfrac{?}{10} \\ &= ? \end{aligned}

Term Plot

Term aⱼ 0

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The 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:

a1,a1+d,a1+2d,a1+3d,…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:

an=a1+(n−1)⋅da_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:

a20=3+(20−1)×4=3+76=79a_{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:

Sn=n⋅(a1+an)2S_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=101,2+99=101,3+98=101,…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:

Sn=n⋅(2a1+(n−1)d)2S_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:

aˉ=Snn=a1+an2\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=a1+(x−1)⋅dy = 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

ConditionEffect
d = 0Constant sequence; every term equals a₁; Sₙ = n·a₁
d < 0Decreasing sequence; terms eventually become negative
d fractionalPerfectly valid — terms are equally spaced decimals
n = 1aₙ = 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.

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