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.

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[Metamathematics]