10.4 The nth term of a quadratic sequence — Learn

Prerequisites: square numbers, linear sequences, and collecting like terms.

Objectives: use constant second differences to find a quadratic nth term. This section develops the Higher content in A25.

First and second differences

The first differences are the changes between consecutive terms. The second differences are the changes between consecutive first differences.

For a quadratic sequence, the second differences are constant. That constant is twice the coefficient of $n^2$.

Example 1. Find the nth term of $2,7,16,29,\ldots$.

First differences are $5,9,13$. Second differences are $4,4$. So start with $2n^2$.

Subtract these terms from the given terms:

The differences follow $1-n$, so the rule is $2n^2-n+1$.

Check at position $4$:

Progress check. Pattern A has constant first differences, so it is linear. Pattern B doubles, so its terms do not have constant second differences.

Watch out. The constant second difference is not the coefficient of $n^2$. Divide it by $2$, then check your rule against the original terms.

You should now be able to

• calculate first and second differences;
• halve the constant second difference to find the $n^2$ coefficient;
• find the remaining linear and constant parts;
• verify a quadratic rule using the first terms.