Explicit rules

An explicit rule gives the nth term straight from n. For the odd numbers, uₙ = 2n – 1: u₁ = 1, u₂ = 3, u₃ = 5, ….

It finds far-off terms at once. The 53rd odd number is 2 × 53 – 1 = 105, and the 1170th is 2339, without listing the ones before.

It also tests whether a number belongs. Is 137 in the list, and where? Solve 2n – 1 = 137: n = 69, so it is the 69th term.

The position must be a natural number. For sₙ = 5n – 2:

• 5n – 2 = 308 gives n = 62, so 308 is a term;
• 5n – 2 = 471 gives n = 94.6. There is no 94.6th position, so 471 is not a term.

Not every sequence has a simple rule. The squares have tₙ = n². The primes 2, 3, 5, 7, 11, … have no known simple rule: the gaps between them keep changing.