Question

How do find the `n`th term in a sequence?

Answer 1

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}`