How do you distinguish an arithmetic sequence or series from other sequences or series?

An arithmetic sequence or series is distinguished from other sequences or series by its constant difference between consecutive terms.

Understanding Arithmetic Sequences and Series

An arithmetic sequence is a list of numbers where the difference between any two successive members is a constant. This constant difference is called the common difference. An arithmetic series is simply the sum of the terms in an arithmetic sequence.

Key Distinguishing Features

The primary way to identify an arithmetic sequence is to check if the difference between consecutive terms is constant. Here’s a breakdown:

• Constant Difference: If subtracting any term from its subsequent term yields the same value, then it's an arithmetic sequence.

Identifying Arithmetic Sequences

Here's a step-by-step approach to determine if a sequence is arithmetic:

1. Calculate the difference: Subtract each term from the term that follows it.
2. Check for consistency: If all the differences calculated in step 1 are the same, the sequence is arithmetic.

Examples

Let's look at some examples to illustrate this:

• Arithmetic Sequence: 2, 4, 6, 8, 10...

  • The difference between consecutive terms is consistently 2 (4-2 = 2, 6-4 = 2, and so on).

• Non-Arithmetic Sequence: 1, 3, 6, 10, 15...

  • The difference between consecutive terms is not constant (3-1 = 2, 6-3 = 3, 10-6 = 4...). Therefore, it's not an arithmetic sequence.

General Term Formula

According to the reference, the general term of an arithmetic sequence can be written as:

• an = a1 + (n - 1)d

Where:

• an is the nth term.
• a1 is the first term.
• n is the term number (index).
• d is the common difference.

This formula is useful for finding any term in the sequence if you know the first term and the common difference.

Table Summarizing the Key Differences