Which Of The Sequences Is An Arithmetic Sequence

How to Identify an Arithmetic Sequence: A Step-by-Step Guide

At the heart of many mathematical patterns lies a simple, elegant rule: a sequence where each term increases or decreases by the exact same amount. This is an arithmetic sequence, a fundamental concept that unlocks understanding in algebra, finance, and science. Recognizing whether a given list of numbers follows this consistent pattern is a crucial skill. This guide will provide you with a clear, methodical approach to determine if any sequence is arithmetic, moving from basic definitions to practical application and real-world significance.

What Exactly Is an Arithmetic Sequence?

An arithmetic sequence (or arithmetic progression) is a ordered list of numbers in which the difference between any two successive members is a constant. This constant is known as the common difference, often denoted by the letter d. The defining characteristic is unwavering consistency in the "step" from one term to the next.

• If d is positive, the sequence increases.
• If d is negative, the sequence decreases.
• If d is zero, the sequence is constant (all terms are identical).

Example of an arithmetic sequence: 5, 9, 13, 17, 21, ... Here, the common difference d = 4 (9 - 5 = 4, 13 - 9 = 4, etc.).

Example of a non-arithmetic sequence: 2, 4, 8, 16, 32, ... The differences are 2, 4, 8, 16—they are not constant. This is a geometric sequence, where each term is multiplied by a common ratio.

How to Identify an Arithmetic Sequence: A Step-by-Step Method

When presented with a sequence, follow this systematic checklist to confirm its nature.

Step 1: Calculate Successive Differences

Subtract each term from the term that immediately follows it. Create a list of these differences.

• For sequence A: 3, 7, 11, 15 → Differences: 4, 4, 4. Constant.
• For sequence B: 10, 7, 4, 1, -2 → Differences: -3, -3, -3, -3. Constant.
• For sequence C: 1, 4, 9, 16 → Differences: 3, 5, 7. Not constant.

If all calculated differences are identical, you have found your common difference (d), and the sequence is arithmetic.

Step 2: Verify with the General Formula

An arithmetic sequence can be described by the explicit formula: aₙ = a₁ + (n - 1)d Where:

• aₙ is the nth term (the term you want to find).
• a₁ is the first term.
• n is the term position (1, 2, 3, ...).
• d is the common difference.

Use this formula to test your findings. For sequence A (3, 7, 11, 15): a₁ = 3, d = 4. The 4th term should be: a₄ = 3 + (4-1)*4 = 3 + 12 = 15. It matches. This double-check confirms the pattern holds for all positions.

Step 3: Look for the Linear Pattern

Plotting the terms of an arithmetic sequence (term number n on the x-axis, term value aₙ on the y-axis) always produces a straight line. This is because the relationship between n and aₙ is linear. The common difference d is the slope of that line. If you can intuitively see that the numbers are rising or falling by the same amount each step, you are recognizing this linearity.

The Formula: Your Mathematical Compass

Understanding the formula aₙ = a₁ + (n - 1)d is key to mastering arithmetic sequences.

• It allows you to find any term without listing all preceding ones. For example, what is the 50th term of the sequence starting at 2 with d =