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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.

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 \(n^2 + 1\).

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 \(2n^2 + 3\).

In a geometric sequence, the term to term rule is to multiply or divide by the same value. This value is called the common ratio, \(r\), which can be worked out by dividing one term by the previous term.

Dividing each term by the previous term gives the same value: \(6 \div 3 = 12 \div 6 = 24 \div 12 = 2\). 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 \(1.4406 \times 0.7 =1.00842\). The common ratio is less than 1. If the size of the common ratio is less than 1, the terms of the sequence will reduce in size.

b) The common ratio is \(3\sqrt{3} \div3 = \sqrt{3}\). The next three terms are \(27 \times \sqrt{3} = 27\sqrt{3}\), \(27\sqrt{3} \times \sqrt{3} = 27 \times 3 = 81,\) and \(81 \times \sqrt{3} = 81\sqrt{3}\). Some of the terms of this sequence are surds, so leave your answer in surds as this is more accurate than writing them in decimal form as they would have to be rounded.

c) The common ratio is \(-4 \div 2 = -2\). The next three terms of the sequence are \(-16 \times -2 = 32\), \(32 \times -2 = -64\), and \(-64 \times -2 = 128\). The common ratio is negative which means that the terms of the sequence will alternate between positive and negative.

The nth term of a geometric sequence is \(ar^{n-1}\), where \(a\) is the first term and \(r\) is the common ratio.

The first term is 2, so \(a = 2\).

The common ratio is \(2.4 \div 2 = 1.2\), so \(r = 1.2\).

The nth term of the geometric sequence is \(ar^{n-1} = 2 \times 1.2^{n-1}\)

The 10th term is \(2 \times 1.2^9= 10.3195607\)