How do you find the general term for a sequence?

1 Answer
Jul 18, 2015

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