Geometric sequences change from one term to the other by multiplication of a constant amount. This amount is also called the common ratio. This common ratio is deduced by dividing any term in the sequence before it.
8, 12, 18, 27……
The common ratio r in this sequence is worked out by:
r = 12 ÷ 8 = 1.5
The nth term of a geometric sequence
In the following example, it is required to find the nth term of the following sequence:
3, 6, 12, 24…….
In this sequence, the first term is 3 and moves on to the next sequence by multiplying by 2. This therefore can be written as:
u1 = 3
u2 = 6 = (3 x 2)
u3 = 12 = (3 x 2 x 2) = 3 x 2²
Therefore this sequence can be written as:
Therefore the nth term of a geometric sequence with the first term as a common ratio r is written as: