How do find the nth term in a sequence?

1 Answer
May 28, 2015

It depends on the type of sequence.

If the sequence is an arithmetic progression with first term a_1, then the terms will be of the form:

a_n = a_1 + (n-1)b
for some constant b.

If the sequence is a geometric progression with first term a_1, then the terms will be of the form:

a_n = a_1 * r^(n-1)
for some constant r.

There are also sequences where the next number is defined iteratively in terms of the previous 2 or more terms. An example of this would be the Fibonacci sequence:

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89,...

Each term is the sum of the two previous terms.

The ratio of successive pairs of terms tends towards the golden ratio phi = 1/2 + sqrt(5)/2 ~= 1.618034

The terms of the Fibonacci sequence are expressible by the formula:

F_n = (phi^n-(-phi)^-n)/sqrt(5) (starting with F_0 = 0, F_1 = 1)

In general an infinite sequence is any mapping from NN -> S for any set S. It can be defined in any way you like.

Finite sequences are the same, except that they are mappings from a finite subset of NN consisting of those numbers less than some fixed limit, e.g. {n in NN: n <= 10}