Arithmetic Sequence
An arithmetic sequence (also called an arithmetic progression) is an ordered list of numbers in which the difference between consecutive terms is constant, called the common difference d. The nth term is given by aₙ = a₁ + (n−1)d, where a₁ is the first term. Arithmetic sequences model uniform increments such as salary increases, regular savings, and equally spaced physical measurements.
Key Formula
a_n = a_1 + (n - 1)d
LaTeX: a_n = a_1 + (n - 1)d
| Symbol | Meaning | Unit |
|---|---|---|
| aₙ | nth term of the sequence | dimensionless |
| a₁ | First term | dimensionless |
| n | Term number (positive integer) | dimensionless |
| d | Common difference (aₙ − aₙ₋₁) | dimensionless |
Worked Example
Problem
An arithmetic sequence has first term 7 and common difference 5. Find the 15th term and the sum of the first 15 terms.
Solution
Step 1: 15th term: a₁₅ = 7 + (15−1)×5 = 7 + 70 = 77. Step 2: Sum of first 15 terms: S₁₅ = (15/2)(a₁ + a₁₅) = (15/2)(7 + 77) = (15/2)(84) = 630.
Answer
a₁₅ = 77; S₁₅ = 630.
Examples of Arithmetic Sequences
| Sequence | First Term (a₁) | Common Difference (d) | Pattern |
|---|---|---|---|
| 2, 5, 8, 11, 14, … | 2 | 3 | Increasing |
| 20, 15, 10, 5, 0, … | 20 | −5 | Decreasing |
| 0, 0.5, 1, 1.5, 2, … | 0 | 0.5 | Fractional increment |
| −3, −1, 1, 3, 5, … | −3 | 2 | Negative to positive |
| 100, 100, 100, … | 100 | 0 | Constant (d = 0) |
Interactive Tools
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Related Terms
Geometric Sequence
A geometric sequence (or geometric progression) is an ordered list of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio r. The nth term is given by aₙ = a₁ × r^(n−1). Geometric sequences model exponential growth and decay, compound interest, population doubling, and signal attenuation.
Mathematical Series
A mathematical series is the sum of the terms of a sequence, either finite or infinite. For a finite series the sum is always a specific number, while an infinite series converges to a finite limit only if its terms decrease fast enough. Series are foundational in calculus, number theory, physics, and engineering — from computing π to modelling oscillations and signal processing.
Linear Equation
A linear equation is an algebraic equation in which each term is either a constant or the product of a constant and a single variable raised to the first power, producing a straight-line graph when plotted. The standard form of a linear equation in one variable is ax + b = 0, while in two variables it is ax + by = c. Linear equations are foundational in algebra and appear throughout science, economics, and engineering for modelling proportional relationships.
From Greek "arithmetike" (the art of counting) and Latin "sequentia" (a following). The systematic study of arithmetic progressions dates to ancient Greece, with contributions from Pythagoras and later formalised by Euclid.