Given several terms in a sequence, it is sometimes possible to find a formula for the general term of the sequence. Such a formula will produce the $n$th term when a value for the integer $n$ is put into the formula.

If a sequence is generated by a polynomial, this fact can be detected by noticing whether the computed differences eventually become constant.

Linear Polynomials

Consider the sequence:

 $5, 7, 9, 11, 13, \dots$

The difference between $7$ and $5$ is $2$. The difference between $7$ and $9$ is also $2$. In fact, the difference between each pair of terms is $2$. Since this difference is constant, and this is the first set of differences, the sequence is given by a first-degree (linear) polynomial. 

Suppose the formula for the sequence is given by $an+b$ for some constants $a$ and $b$. Then the sequence looks like:

$a+b, 2a+b, 3a+b, \dots$

The difference between each term and the term after it is $a$. In our example, $a=2$. It is possible to solve for $b$ using one of the terms in the sequence. Using the first number in the sequence and the first term:

$\displaystyle{ \begin{aligned} 5 &= a+b \\ b &= 5-a \\ b &= 5-(2)\\ b &= 3 \end{aligned} }$

So, the $n$th term of the sequence is given by $2n+3$.

Quadratic Polynomials

Suppose the $n$th term of a sequence was given by $an^2+bn+c$. Then the sequence would look like:

$a+b+c, 4a+2b+c, 9a+3b+c, \dots$

This sequence was created by plugging in $1$ for $n$, $2$ for $n$, $3$ for $n$, etc.

If we start at the second term, and subtract the previous term from each term in the sequence, we can get a new sequence made up of the differences between terms. The first sequence of differences would be:

$3a+b, 5a+b, 7a+b, \dots$

Now, we take the differences between terms in the new sequence. The second sequence of differences is:

$2a, 2a, 2a, 2a, \dots$

The computed differences have converged to a constant after the second sequence of differences. This means that it was a second-order (quadratic) sequence. Working backward from this, we could find the general term for any quadratic sequence.

Example

Consider the sequence:

$4, -7, -26, -53, -88, -131, \dots$

The difference between $-7$ and $4$ is $-11$, and the difference between $-26$ and $-7$ is $-19$. Finding all these differences, we get a new sequence:

$-11, -19, -27, -35, -43, \dots$

This list is still not constant. However, finding the difference between terms once more, we get:

$-8, -8, -8, -8, \dots$

This fact tells us that there is a polynomial formula describing our sequence. Since we had to do differences twice, it is a second-degree (quadratic) polynomial.

We can find the formula by realizing that the constant term is $-8$, and that it can also be expressed by $2a$. Therefore $a=-4$. 

Next we note that the first item in our first list of differences is $-11$, but that generically it is supposed to be $3a+b$, so we must have $3(-4)+b=-11$, and $b=1$.

Finally, note that the first term in the sequence is $4$, and can also be expressed by 

$a+b+c = -4+1+c$

So, $c=7$, and the formula that generates the sequence is $-4a^2+b+7c$.

General Polynomial Sequences

This method of finding differences can be extended to find the general term of a polynomial sequence of any order. For higher orders, it will take more rounds of taking differences for the differences to become constant, and more back-substitution will be necessary in order to solve for the general term.

General Terms of Non-Polynomial Sequences

Some sequences are generated by a general term which is not a polynomial. For example, the geometric sequence $2, 4, 8, 16,\dots$ is given by the general term $2^n$. Because this term is not a polynomial, taking differences will never result in a constant difference.

General terms of non-polynomial sequences can be found by observation, as above, or by other means which are beyond our scope for now. Given any general term, the sequence can be generated by plugging in successive values of $n$.