Geometric sequence

Let’s say we have a sequence of numbers 5, 10, 20, 40, 80,…

How can we describe this sequence?

The sequence starts with 5

The ratio of a number in the sequence and the previous number in the sequence is always 2

It can be useful to put the sequence in a table

If we label each number in the sequence by n and the value of each number with a_n and the ratio between adjacent numbers in the sequence with r then we can put the sequence into a table like this

Now we can write a formula to give the value of any term in the sequence

$a_n = a_1r^{n-1}$

Let’s check this works for the 4th term

$a_4 = a_1r^3$

$a_4 = 5\times2^3 = 5\times8 = 40$

And work out what the 10th term is

$a_{10} = 5\times2^{9} = 5\times512 = 2560$

This type of sequence is called geometric because it can be shown with geometric shapes. In the example that we have looked at, the ratio between successive terms is 2.

Here is a geometric representation of the first 5 terms of the sequence. The sides of the squares increase in a geometric sequence.

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Numbers Part Two

Geometric sequence