Arithmetic Sequences: The Complete Guide to Patterns and Progressions

An arithmetic sequence is one of the most fundamental patterns in mathematics — a sequence of numbers where the difference between consecutive terms is constant. From savings plans to scheduling, arithmetic sequences appear everywhere in daily life and form the foundation for more advanced mathematical concepts.

What Is an Arithmetic Sequence?

An arithmetic sequence (also called an arithmetic progression) is an ordered list of numbers where each term after the first is obtained by adding a fixed constant called the common difference (denoted as d).

For example, the sequence 3, 7, 11, 15, 19, ... has a common difference of 4. Each term is exactly 4 more than the previous term.

The nth Term Formula

Where a₁ is the first term, d is the common difference, and n is the term number. This formula lets you find any term directly without computing all preceding terms.

The Sum Formula

The sum of the first n terms can be calculated using either form. The second form, using the first and last terms, is particularly elegant — it was famously discovered by young Carl Friedrich Gauss.

Worked Example

Find the 20th term and the sum of the first 20 terms of the sequence 5, 11, 17, 23, ...:

1. Identify: a₁ = 5, d = 6, n = 20
2. 20th term: a₂₀ = 5 + (20-1)(6) = 5 + 114 = 119
3. Sum: S₂₀ = 20/2 × (2×5 + 19×6) = 10 × (10 + 114) = 10 × 124 = 1240

Real-World Applications

• Savings plans: Depositing $100, $200, $300, ... each month
• Salary schedules: Annual raises of a fixed amount
• Theater seating: Rows with equal increases in seat count
• Physics: Uniform acceleration produces arithmetic velocity sequences

Key Properties

• The average of the first and last terms equals the average of all terms
• If d > 0, the sequence is increasing; if d < 0, it is decreasing
• If d = 0, all terms are equal (constant sequence)
• The graph of aₙ vs. n is a straight line with slope d