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Quadratic sequences are sequences that include an \(n^2\) term. They can be identified by the fact that the differences between the terms are not equal, but the second differences between terms are equal.

The \(n\)th term for a quadratic sequence has a term that contains \(n^2\). Terms of a quadratic sequence can be worked out in the same way.

Write the first five terms of the sequence \(n^2 + 3n - 5\).

• when \(n = 1\), \(n^2 + 3n - 5 = 1^2 + 3 \times 1 - 5 = 1 + 3 – 5 = -1\)
• when \(n = 2\), \(n^2 + 3n - 5 = 2^2 + 3 \times 2 - 5 = 4 + 6 – 5 = 5\)
• when \(n = 3\), \(n^2 + 3n - 5 = 3^2 + 3 \times 3 - 5 = 9 + 9 – 5 = 13\)
• when \(n = 4\), \(n^2 + 3n - 5 = 4^2 + 3 \times 4 - 5 = 16 + 12 – 5 = 23\)
• when \(n = 5\), \(n^2 + 3n - 5 = 5^2 + 3 \times 5 - 5 = 25 + 15 – 5 = 35\)

The first five terms of the sequence: \(n^2 + 3n - 5\) are -1, 5, 13, 23, 35

Work out the \(nth\) term of the sequence 2, 5, 10, 17, 26, ...

The second differences are the same. The sequence is quadratic and will contain an \(n^2\) term. The coefficient 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 \(n\)th term of this sequence is therefore \(n^2 + 1\) .

Work out the \(n\)th term of the sequence 5, 11, 21, 35, ...

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 \(2n^2 + 3\).

The coefficient of \(n^2\) is half the second difference, which is 1.

So 8, 13, 18, 23 is a linear sequence with \(n\)th term \(5n + 3\).

So the \(n\)th term of the quadratic sequence is \(n^2 + 5n + 3\).