forms of the n-th number in a sequence

The n-th number in a given sequence of numbers might have more than one commonly‑encountered form.

In the case of a sequence of equally‑spaced numbers, there are the following two forms that are commonly encountered:

first form: a + (n – 1)d

second form: (a – d) + nd

They are algebraically equivalent, of course. The first form is the ‘definition’ form, while the second form is the ‘business’ form, the form more likely to be encountered in applications, and on a test.



thematic summation

Sums are usually of arbitrary numbers, but sometimes a sum has a theme. A good example of this concerns the odd numbers, namely, the sum of the first n odd numbers. This thematic summation is of interest because of its gem of an answer: ‘n squared’. For example, the first 3 odd numbers are 1, 3, and 5, and their sum is 9, which is 3 squared.



[Number Theory]