Arithmetic Sequence

Arithmetic Sequence
Arithmetic Progression

A sequence such as 1, 5, 9, 13, 17 or 12, 7, 2, –3, –8, –13, –18 which has a constant difference between terms. The first term is a₁, the common difference is d, and the number of terms is n.

Explicit Formula:	a_n = a₁ + (n – 1)d
Example 1:	3, 7, 11, 15, 19 has a₁ = 3, d = 4, and n = 5. The explicit formula is a_n = 3 + (n – 1)·4 = 4n – 1
Example 2:	3, –2, –7, –12 has a₁ = 3, d = –5, and n = 4. The explicit formula is a_n = 3 + (n – 1)(–5) = 8 – 5n

See also

Arithmetic series, geometric sequence

Key Formula

a_n = a_1 + (n - 1)d

Where:

$a_n$ = The nth term of the sequence (the term you want to find)
$a_1$ = The first term of the sequence
$n$ = The position number of the term (must be a positive integer)
$d$ = The common difference between consecutive terms

Worked Example

Problem: Find the 20th term of the arithmetic sequence 6, 10, 14, 18, …

Step 1: Identify the first term.

a_1 = 6

Step 2: Find the common difference by subtracting any term from the next term.

d = 10 - 6 = 4

Step 3: Write the explicit formula with the known values.

a_n = 6 + (n - 1)(4)

Step 4: Substitute n = 20 into the formula.

a_{20} = 6 + (20 - 1)(4) = 6 + 19 \cdot 4

Step 5: Simplify to get the answer.

a_{20} = 6 + 76 = 82

Answer: The 20th term of the sequence is 82.

Another Example

This example works backward from two known terms to find the first term and common difference, rather than simply plugging values into the formula.

Problem: The 5th term of an arithmetic sequence is 23 and the 12th term is 58. Find the first term and the common difference.

Step 1: Write the explicit formula for each known term.

a_5 = a_1 + 4d = 23 \quad \text{and} \quad a_{12} = a_1 + 11d = 58

Step 2: Subtract the first equation from the second to eliminate the first term.

(a_1 + 11d) - (a_1 + 4d) = 58 - 23 \implies 7d = 35

Step 3: Solve for the common difference.

d = \frac{35}{7} = 5

Step 4: Substitute d = 5 back into either equation to find the first term.

a_1 + 4(5) = 23 \implies a_1 = 23 - 20 = 3

Answer: The first term is 3 and the common difference is 5. The sequence begins 3, 8, 13, 18, 23, …

Frequently Asked Questions

What is the difference between an arithmetic sequence and an arithmetic series?

An arithmetic sequence is the ordered list of terms (e.g., 2, 5, 8, 11), while an arithmetic series is the sum of those terms (e.g., 2 + 5 + 8 + 11 = 26). The sequence focuses on the individual terms; the series focuses on their total.

Can the common difference in an arithmetic sequence be negative or zero?

Yes. A negative common difference produces a decreasing sequence, such as 20, 15, 10, 5 where d = −5. A common difference of zero gives a constant sequence like 4, 4, 4, 4, which is technically arithmetic but not very interesting.

How do you find the common difference of an arithmetic sequence?

Subtract any term from the term that immediately follows it: d = a_{n+1} − a_n. For the sequence 7, 11, 15, 19, the common difference is 11 − 7 = 4. This value should be the same for every pair of consecutive terms; if it's not, the sequence is not arithmetic.

Arithmetic Sequence vs. Geometric Sequence

	Arithmetic Sequence	Geometric Sequence
Pattern	Add a constant difference between terms	Multiply by a constant ratio between terms
Explicit formula	a_n = a_1 + (n − 1)d	a_n = a_1 · r^(n − 1)
Example	2, 5, 8, 11, 14 (d = 3)	2, 6, 18, 54, 162 (r = 3)
Growth behavior	Linear — graph forms a straight line	Exponential — graph curves steeply up or down
When it equals zero	Terms can be zero and continue past it	If any term is zero, all following terms are zero

Why It Matters

Arithmetic sequences appear throughout algebra, precalculus, and standardized tests like the SAT and ACT. They model real-world situations with constant rates of change, such as saving a fixed amount of money each month or counting seats in an auditorium where each row has two more seats than the last. Understanding them also lays the groundwork for arithmetic series, linear functions, and later topics in calculus.

Common Mistakes

Mistake: Using n instead of (n − 1) in the formula, writing a_n = a_1 + nd.

Correction: The formula is a_n = a_1 + (n − 1)d because the first term has zero differences added. For example, the 1st term is a_1 + (1 − 1)d = a_1, not a_1 + d.

Mistake: Forgetting that the common difference can be negative, and computing d as a positive number when the sequence is decreasing.

Correction: Always subtract in the correct order: d = (next term) − (current term). For the sequence 20, 14, 8, 2, you get d = 14 − 20 = −6, not 6.

Related Terms

Sequence — General concept that arithmetic sequences are a type of
Arithmetic Series — The sum of an arithmetic sequence's terms
Geometric Sequence — Sequence defined by a constant ratio instead of difference
Explicit Formula of a Sequence — Formula giving any term directly from its position
Constant — The common difference is a constant value
Difference — The subtraction operation that defines d
Term — Each individual element in the sequence