###### The Geometric Sequence

Maths Applications (Year 12) - Growth and Decay in Sequences

Bhakti Sharma

Geometric Sequences are those in which each value, after the first, is obtained by the product of the previous term with a constant amount, or a common ratio. Another name for geometric sequences is geometric progressions (GP). Letâ€™s look at an example:

In the table above, the x-values increase by 1, and the y-values increase by multiplying the previous term by 2. If we were to graph these values, we would obtain an exponential graph, as shown below:

Letâ€™s look at some more examples of GPs:

4, 8, 16, 32, 64, 128

In this sequence, the first term is 4 and the common ratio is 2

10, 50, 250, 1250, 6250

In this sequence, the first term is 10 and the common ratio is 5

5000, 2500, 1250, 625, 312.5

In this sequence, the first term is 5000 and the common ratio is 0.5

Following this, we can define the form of GPs as:

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With â€˜*a*â€™ being the first term, and â€˜*r*â€™ being the common term.

If we were to use recursive notation, we would use:

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Letâ€™s suppose that we wanted to jump to an *nth* term in the geometric sequence. We could do this by the following formula:

With *T1* as the first term and *r* as the common term.