How to find the common ratio of a geometric sequence

Finding the Common Ratio of a Geometric Sequence

Geometric Sequence

Image by: https://pixabay.com/photos/perspective-symmetry-pattern-5776865/

To find the common ratio of a geometric sequence, you can divide any term in the sequence by the previous term. The common ratio, usually denoted by r, is consistent throughout the sequence.

If the sequence is a1, a2, a3, …, then the common ratio r is given by:

r = a2 / a1 = a3 / a2 = a4 / a3 = …

Example:

Suppose you have the geometric sequence: 3, 6, 12, 24, ….

  1. Take the second term 6 and divide it by the first term 3: 
    r = 6 / 3 = 2
  2. You can check it by dividing the third term 12 by the second term 6: 
    r = 12 / 6 = 2

The common ratio r for this sequence is 2.

Finding the Common Ratio When the Sequence Oscillates Between Positive and Negative

If a geometric sequence oscillates between positive and negative numbers, the process for finding the common ratio is the same. The difference is that the common ratio r will be a negative number. You still divide any term by the previous term, and if the sequence alternates between positive and negative, r will reflect that sign change.

Example:

Consider the sequence: 5, -10, 20, -40, ….

  1. Take the second term -10 and divide it by the first term 5: 
    r = -10 / 5 = -2
  2. To confirm, divide the third term 20 by the second term -10: 
    r = 20 / -10 = -2

The common ratio r is -2, indicating that each term is obtained by multiplying the previous term by -2. This negative ratio causes the sequence to alternate between positive and negative values.

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