Consider all the sequences we have discussed so far in this chapter. Which ones are arithmetic progressions and which ones are not? Can you justify your claim?
Among the sequences discussed, the arithmetic progressions (APs) are: 1, 5, 9, 13, 17, 21; 1, 4, 7, 10, …; 11, 7, 3, −1, −5, …; 2, 5, 8, 11, …; −5, −1, 3, 7, …; 1/2, 5/2, 9/2, 13/2, …; and 1.5, 3.5, 5.5, 7.5, …
The sequences that are not APs include the sequence of prime numbers, geometric progressions like 2, 10, 50, 250, … and 3, 30, 300, 3000, …, and the recursive sequence 1, 2, 4, 7, 13, ….
I justify this using the definition of an AP. An arithmetic progression has a constant common difference between consecutive terms, expressible as . The first group satisfies this (e.g., in 1, 5, 9, 13..., the difference is 4). Non-APs lack this constant difference; geometric progressions have a constant ratio, the prime number sequence lacks regularity, and the recursive sequence has varying differences.
Explanation
The textbook introduces arithmetic progressions as sequences where the difference between consecutive terms is constant, given by the formula t_n = a + (n-1)d. The sequences derived from the growing pattern of squares (1, 5, 9, 13...) and the exercise examples (2, 5, 8, 11..., etc.) all share this property of a constant common difference. Conversely, sequences like prime numbers lack clear regularity, geometric progressions rely on a constant ratio rather than difference, and recursive sequences defined by sums of previous terms do not yield constant differences, thus they are not APs.