Fundamentals of Arithmetic Progression

Definition and Fundamentals of Arithmetic Progression (AP)

An Arithmetic Progression, commonly referred to by the acronym AP, is defined as a sequence of numbers in which the difference between any two consecutive terms is consistently identical throughout the sequence. This characteristic fixed difference is formally designated as the common difference, represented by the variable $d$. In an Arithmetic Progression, each term is generated by adding this common difference to the preceding term, establishing a linear and predictable pattern throughout the numerical string.

Illustrative Examples of Arithmetic Progressions

The transcript provides specific examples to demonstrate both increasing and decreasing sequences. The first example given is the sequence $2, 5, 8, 11, \dots$. This sequence is classified as an Arithmetic Progression because every subsequent term undergoes a uniform increase of $3$. Consequently, the common difference ($d$) for this sequence is $3$. A second example provided is $20, 17, 14, 11, \dots$. This is also a valid Arithmetic Progression, as each successive term decreases by a fixed value of $3$. In this case, the common difference ($d$) is $-3$, illustrating that the common difference can be either positive or negative depending on whether the sequence is increasing or decreasing.

Verification of an Arithmetic Progression

To identify or verify whether a particular sequence of numbers constitutes an Arithmetic Progression, a systematic subtraction of consecutive terms must be performed. The process involves taking a term and subtracting the term that immediately precedes it. To confirm the sequence is an AP, this calculation must be repeated for multiple pairs of consecutive terms throughout the sequence. If the resulting difference remains constant across all instances, the sequence is confirmed to be an Arithmetic Progression. If the difference varies at any point, the sequence does not satisfy the criteria for an AP.