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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). Let’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:

Let’s look at some more examples of GPs:

  1. 4, 8, 16, 32, 64, 128

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

  2. 10, 50, 250, 1250, 6250

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

  3. 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:

 

With ‘a’ being the first term, and ‘r’ being the common term.

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

 

Let’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.

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