Quadratic sequences
Quadratic sequences are sequences that include an \(n^2\) term. They can be identified by the fact that the differences in between the terms are not equal, but the second differences between terms are equal.
Example one
Work out the \(n\)th term of the sequence 2, 5, 10, 17, 26, ...
Work out the first differences between the terms. The first differences are not the same, so work out the second differences.
The second differences are the same. The sequence is quadratic and will contain an \(n^2\) term. The The amount that a letter has been multiplied by. In the example of 3a, the coefficient of a is 3 because 3 x a = 3a. of \(n^2\) is always half of the second difference. In this example, the second difference is 2. Half of 2 is 1, so the coefficient of \(n^2\) is 1.
To work out the \(n\)th term of the sequence, write out the numbers in the sequence \(n^2\) and compare this sequence with the sequence in the question.
In this example, you need to add \(1\) to \(n^2\) to match the sequence. The sequence is therefore \(n^2 + 1\).
Example two
Work out the \(n\)th term of the sequence 5, 11, 21, 35, ...
Work out the first differences between the terms. The first differences are not the same, so work out the second differences.
The second difference is the same so the sequence is quadratic and will contain an \(n^2\) term. The coefficient of \(n^2\) is half the second difference, which is 2. The sequence will contain \(2n^2\), so use this:
The sequence is \(2n^2 + 3\).
More guides on this topic
- Equations of lines – WJEC
- Equations of curves - Intermediate & Higher tier – WJEC
- Basic algebra – WJEC
- Equations and formulae – WJEC
- Factorising - Intermediate & Higher tier – WJEC
- Quadratic expressions - Intermediate & Higher tier – WJEC
- Functions - Higher only – WJEC
- Inequalities - Intermediate & Higher tier – WJEC
- Simultaneous equations - Intermediate & Higher tier – WJEC