A Theorem on Arithmetic Progression and Divisibility
A Theorem on Arithmetic Progression and Divisibility In this post, we briefly discuss a particular theorem concerning the divisibility of a sequence of terms in arithmetic progression. Before that, let's take a quick review of an arithmetic progression. Arithmetic Progression: A sequence of numbers that have the same common difference between consecutive terms is said to be an arithmetic progression. For example, ` 1, 5, 9, 15, 19, ... ` is a sequence of numbers that have a common difference of `4`. Mark that is not the absolute difference but rather algebraic difference. Thus the above is said to be an increasing AP with common difference `4`. The common difference is taken algebraically as ( any term) - (the previous term). More examples: ` \frac{1}{2} , 1, \frac{3}{2}, 2, \frac{5}{2}, ... ` ` \sqrt{10}, \sqrt{10} - \sqrt{2}, \sqrt{10} - 2\sqrt{2}, ... ` See also that any series of consecutive terms taken from an arithmetic progression also form an arithmetic ...