# How do find the nth term in a sequence?

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:

${a}_{n} = {a}_{1} + \left(n - 1\right) b$
for some constant b.

If the sequence is a geometric progression with first term ${a}_{1}$, then the terms will be of the form:

${a}_{n} = {a}_{1} \cdot {r}^{n - 1}$
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:

$0 , 1 , 1 , 2 , 3 , 5 , 8 , 13 , 21 , 34 , 55 , 89 , \ldots$

Each term is the sum of the two previous terms.

The ratio of successive pairs of terms tends towards the golden ratio $\phi = \frac{1}{2} + \frac{\sqrt{5}}{2} \cong 1.618034$

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

${F}_{n} = \frac{{\phi}^{n} - {\left(- \phi\right)}^{-} n}{\sqrt{5}}$ (starting with ${F}_{0} = 0$, ${F}_{1} = 1$)

In general an infinite sequence is any mapping from $\mathbb{N} \to S$ 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 $\mathbb{N}$ consisting of those numbers less than some fixed limit, e.g. $\left\{n \in \mathbb{N} : n \le 10\right\}$