Arithmetic Sequence Formula

The Formula

a_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 = 4; the nth term is 3 + (n-1) \cdot 4.

Notation

d = common difference, a_1 = first term, S_n = \frac{n}{2}(a_1 + a_n) = sum of first n terms.

What This Formula Means

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

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

Formal View

A sequence (a_n) is arithmetic if \exists d \in \mathbb{R} : a_{n+1} - a_n = d for all n \geq 1. General term: a_n = a_1 + (n-1)d. Partial sum: 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 a_1 = 7 and d = -3. Find a_{20} and S_{20}.

Solution

  1. 1
    Use the arithmetic sequence formula to find the 20th term: a_n = a_1 + (n-1)d, where a_1 = 7, d = -3, n = 20.
  2. 2
    Calculate: a_{20} = 7 + (20-1)(-3) = 7 - 57 = -50
  3. 3
    Apply the partial sum formula: S_{20} = \frac{20}{2}(a_1 + a_{20}) = 10(7 + (-50)) = 10(-43) = -430

Answer

a_{20} = -50; S_{20} = -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 a_5 = 18 and a_{12} = 46. Find a_1 and d.

Example 3

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

Common Mistakes

  • Using (n) instead of (n-1) in the formula: the nth term is a_n = a_1 + (n-1)d, not a_1 + nd — off-by-one errors are extremely common.
  • Confusing the common difference with the ratio: in an arithmetic sequence you ADD d each time; if you're MULTIPLYING, it's a geometric sequence.
  • Miscounting the number of terms: from a_3 to a_{10}, there are 8 terms (not 7) — count inclusively: 10 - 3 + 1 = 8.

Why This Formula Matters

Arithmetic sequences model any situation with constant rates of change—savings plans, evenly spaced measurements.

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 d.

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?

d = common difference, a_1 = first term, S_n = \frac{n}{2}(a_1 + a_n) = sum of first n terms.

Why is the Arithmetic Sequence formula important in Math?

Arithmetic sequences model any situation with constant rates of change—savings plans, evenly spaced measurements.

What do students get wrong about Arithmetic Sequence?

Use a_n = a_1 + (n-1)d, not a_1 + nd — off-by-one errors are very common. Sum: S_n = \frac{n(a_1 + a_n)}{2}.

What should I learn before the Arithmetic Sequence formula?

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

Want the Full Guide?

This formula is covered in depth in our complete guide:

Growing Patterns, Arithmetic and Geometric Sequences →
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