Geometric sequences

In a \(geometric\) sequence, the term to term rule is to multiply or divide by the same value.

Example

Show that the sequence 3, 6, 12, 24, … is a geometric sequence, and find the next three terms.

Dividing each term by the previous term gives the same value: \(\frac{6}{3} = \frac{12}{6} = \frac{24}{12} = 2\).

Each term is double the previous term so the sequence is geometric

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

Example

Find the next three terms in the geometric sequences:

a) 6, 4.2, 2.94, ...

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

c) 2, -4, 8, -16, ...

a) Working out 4.2 divided by 6 gives 0.7. To find each term, multiply the previous term by 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

\(1.4406 \times 0.7 =1.00842\)

Because you multiply by a number less than 1, the terms are getting smaller.

b) Working out \(3\sqrt{3}\) divided by 3 gives √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) Working out –4 divided by 2 gives –2.

The next three terms of the sequence are \(–16 \times –2 = 32\), \(32 \times –2 = −64\), and \(–64 \times –2 = 128\).

The terms of the sequence will alternate between positive and negative.

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Sequences - AQAGeometric sequences