Arithmetic Sequences - Algebra-1

1. Fundamental Concepts

Arithmetic Sequences: A special type of sequence characterized by the constant difference between consecutive terms. This constant difference is called the common difference.

Judgment Criterion: For a sequence, if the difference between any two consecutive terms (latter term minus former term) is always equal (i.e., there exists a fixed common difference d), then the sequence is an arithmetic sequence; conversely, if the differences between consecutive terms are not constant, it is not an arithmetic sequence.
Role of Common Difference (d): The common difference is a key indicator for judging an arithmetic sequence. It is calculated by the formula \(d = a_{n+1} - a_n\) (where n is a positive integer). A sequence can be determined as an arithmetic sequence as long as this difference remains consistent for all consecutive terms.

The sequence \(2, 5, 8, \dots\): Calculate the differences between consecutive terms: \(5 - 2 = 3\), \(8 - 5 = 3\). The difference is constant (\(d = 3\)), so it is an arithmetic sequence.
A sequence with \(a_1 = 1\) and calculated \(d = -2.5\): Since there is a fixed common difference of \(-2.5\), it meets the definition of an arithmetic sequence and thus is an arithmetic sequence.

Step 1: Select any two consecutive terms in the sequence and calculate the difference (i.e., \(d = a_2 - a_1\), \(d = a_3 - a_2\), etc.).
Step 2: Check if all the calculated differences are exactly the same.
Step 3: If the differences between all consecutive terms are consistent, the sequence is an arithmetic sequence; if there is at least one pair of consecutive terms with a different difference, it is not an arithmetic sequence.