Arithmetic Sequence

Calculus
definition

Also known as: arithmetic progression

Grade 9-12

View on concept map

A sequence where each term is obtained from the previous by adding a fixed constant called the common difference d. Arithmetic sequences model any situation with constant rates of change—savings plans, evenly spaced measurements.

This concept is covered in depth in our arithmetic sequences and common differences, with worked examples, practice problems, and common mistakes.

Definition

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

💡 Intuition

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

🎯 Core Idea

Linear growth—the graph of an arithmetic sequence is a line.

Example

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

Formula

a_n = a_1 + (n-1)d

Notation

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

🌟 Why It Matters

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

💭 Hint When Stuck

Subtract consecutive terms to find d, then verify by checking that the same d works for every pair.

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).

🚧 Common Stuck Point

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}.

⚠️ 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.

Frequently Asked Questions

What is Arithmetic Sequence in Math?

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

Why is Arithmetic Sequence important?

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

What do students usually 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 Arithmetic Sequence?

Before studying Arithmetic Sequence, you should understand: sequence.

Prerequisites

sequence

How Arithmetic Sequence Connects to Other Ideas

To understand arithmetic sequence, you should first be comfortable with sequence. Once you have a solid grasp of arithmetic sequence, you can move on to geometric sequence.

Want the Full Guide?

This concept is explained step by step in our complete guide:

Growing Patterns, Arithmetic and Geometric Sequences →
← Back to Explorer

Visualization

Static

Visual representation of Arithmetic Sequence