Question

How do you find the general term for a sequence?

Answer 1

It depends.

Explanation:

There are many types of sequences. Some of the interesting ones can be found at the online encyclopedia of integer sequences at https://oeis.org/

Let's look at some simple types:

Arithmetic Sequences

`a_n = a_0 + dn`

e.g. `2, 4, 6, 8,...`

There is a common difference between each pair of terms.

If you find a common difference between each pair of terms, then you can determine `a_0` and `d`, then use the general formula for arithmetic sequences.

Geometric Sequences

`a_n = a_0 * r^n`

e.g. `2, 4, 8, 16,...`

There is a common ratio between each pair of terms.

If you find a common ratio between pairs of terms, then you have a geometric sequence and you should be able to determine `a_0` and `r` so that you can use the general formula for terms of a geometric sequence.

Iterative Sequences

After the initial term or two, the following terms are defined in terms of the preceding ones.

e.g. Fibonacci

`a_0 = 0`
`a_1 = 1`
`a_(n+2) = a_n + a_(n+1)`

For this sequence we find: `a_n = (phi^n - (-phi)^(-n))/sqrt(5)` where `phi = (1+sqrt(5))/2`

There are many ways to make these iterative rules, so there is no universal method to provide an expression for `a_n`

Polynomial Sequences

If the terms of a sequence are given by a polynomial, then given the first few terms of the sequence you can find the polynomial.

e.g.

`color(red)(1), 2, 4, 7, 11,...`

Form the sequence of differences of these values:

`color(red)(1), 2, 3, 4,...`

Form the sequence of differences of these values:

`color(red)(1), 1, 1,...`

Once you reach a constant sequence like this, pick out the initial terms from each sequence. In this case `1`, `1` and `1`.

These form the coefficients of a polynomial expression:

`a_n = color(red)(1)/(0!) + (color(red)(1)*n)/(1!) + (color(red)(1)*n(n-1))/(2!)`

`=n^2/2+n/2+1`