Arithmetic Sequence Formula

Arithmetic sequence is a sequence where each term is obtained from the previous by adding a fixed constant called the common difference d.

The Formula

an=a1+(n−1)da_n = a_1 + (n-1)d

When to use: Add the same number each time — 2, 5, 8, 11,... (add 3 each step). This is constant-rate growth.

Quick Example

3, 7, 11, 15, 19,... — common difference d=4d = 4; the nnth term is 3+(n−1)⋅43 + (n-1) \cdot 4.

Notation

dd = common difference, a1a_1 = first term, Sn=n2(a1+an)S_n = \frac{n}{2}(a_1 + a_n) = sum of first nn terms.

What This Formula Means

A sequence where each term is obtained from the previous by adding a fixed constant called the common difference dd.

Add the same number each time — 2, 5, 8, 11,... (add 3 each step). This is constant-rate growth.

Formal View

A sequence (an)(a_n) is arithmetic if ∃d∈R:an+1−an=d\exists d \in \mathbb{R} : a_{n+1} - a_n = d for all n≥1n \geq 1. General term: an=a1+(n−1)da_n = a_1 + (n-1)d. Partial sum: Sn=∑k=1nak=n2(2a1+(n−1)d)S_n = \sum_{k=1}^{n} a_k = \frac{n}{2}(2a_1 + (n-1)d).

Worked Examples

Example 1

easy
An arithmetic sequence has a1=7a_1 = 7 and d=−3d = -3. Find a20a_{20} and S20S_{20}.

Answer

a20=−50a_{20} = -50; S20=−430S_{20} = -430

First step

1
Use the arithmetic sequence formula to find the 20th term: an=a1+(n−1)da_n = a_1 + (n-1)d, where a1=7a_1 = 7, d=−3d = -3, n=20n = 20.

Full solution

  1. 2
    Calculate: a20=7+(20−1)(−3)=7−57=−50a_{20} = 7 + (20-1)(-3) = 7 - 57 = -50
  2. 3
    Apply the partial sum formula: S20=202(a1+a20)=10(7+(−50))=10(−43)=−430S_{20} = \frac{20}{2}(a_1 + a_{20}) = 10(7 + (-50)) = 10(-43) = -430
With a negative common difference the sequence decreases. The sum formula averages the first and last terms and multiplies by the count.

Example 2

medium
In an arithmetic sequence a5=18a_5 = 18 and a12=46a_{12} = 46. Find a1a_1 and dd.

Example 3

medium
Find the sum of the first 20 terms of the arithmetic sequence: 5, 8, 11, 14, ...

Common Mistakes

  • Using a1+nda_1+nd instead of a1+(n−1)da_1+(n-1)d — the first term already counts, so add dd only (n−1)(n-1) times.
  • Confusing constant difference with constant ratio — subtract neighbors for arithmetic, divide for geometric.
  • Reading dd from the wrong direction — d=an+1−and=a_{n+1}-a_n (later minus earlier); a decreasing sequence has a negative dd.

Why This Formula Matters

Arithmetic sequences model anything with a steady per-step increase — savings of a fixed amount per week, seats added per theater row — and their explicit formula an=a1+(n−1)da_n=a_1+(n-1)d lets you jump to the 100th term without listing all of them. The defining check, constant difference, is what separates them from geometric (constant ratio) growth. Recognizing it by "Do I get the same number every time I subtract a term from the one after it?" — rather than by familiar numbers — is what lets a student tell it apart from geometric sequence and arithmetic series and linear function in a mixed problem set.

Frequently Asked Questions

What is the Arithmetic Sequence formula?

A sequence where each term is obtained from the previous by adding a fixed constant called the common difference dd.

How do you use the Arithmetic Sequence formula?

Add the same number each time — 2, 5, 8, 11,... (add 3 each step). This is constant-rate growth.

What do the symbols mean in the Arithmetic Sequence formula?

dd = common difference, a1a_1 = first term, Sn=n2(a1+an)S_n = \frac{n}{2}(a_1 + a_n) = sum of first nn terms.

Why is the Arithmetic Sequence formula important in Math?

Arithmetic sequences model anything with a steady per-step increase — savings of a fixed amount per week, seats added per theater row — and their explicit formula an=a1+(n−1)da_n=a_1+(n-1)d lets you jump to the 100th term without listing all of them. The defining check, constant difference, is what separates them from geometric (constant ratio) growth. Recognizing it by "Do I get the same number every time I subtract a term from the one after it?" — rather than by familiar numbers — is what lets a student tell it apart from geometric sequence and arithmetic series and linear function in a mixed problem set.

What do students get wrong about Arithmetic Sequence?

The procedure for arithmetic sequence is the easy part; the trap is using a1+nda_1+nd instead of a1+(n−1)da_1+(n-1)d. Asking "Do I get the same number every time I subtract a term from the one after it?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

What should I learn before the Arithmetic Sequence formula?

Before studying the Arithmetic Sequence formula, you should understand: sequence.

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Growing Patterns, Arithmetic and Geometric Sequences →
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