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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 \(nth\) 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 \(n^2 + 1\).

Work out the \(nth\) 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 \(2n^2 + 3\).

Dividing each term by the previous term gives the same value: \(\frac{6}{3} = \frac{12}{6} = \frac{24}{12} = 2\). So the common ratio is 2 and this is therefore a geometric sequence.

The next three terms are: \(24 \times 2 = 48\), \(48 \times 2 = 96\) and \(96 \times 2 = 192\).

b) 3, \(3\sqrt{3}\), 9, \(9\sqrt{3}\), 27, ...

a) To find the value of the common ratio, work out \( 4.2 \div 6 = 0.7\) The next three terms of the sequence are \(2.94 \times 0.7 = 2.058\), \(2.058 \times 0.7 = 1.4406\) and \(1.4406 \times 0.7 =1.00842\)

b) The common ratio is \(3\sqrt{3} \div 3 = \sqrt{3}\)

The next three terms are: \(27 \times \sqrt{3} = 27\sqrt{3}, 27\sqrt{3} \times \sqrt{3} = 27 \times 3 = 81\) and \(81 \times \sqrt{3} = 81\sqrt{3}\).