Find the common ratio of the geometric sequence with terms one over 156, one over 52, three over 52, nine over 52, and 27 over 52.
The common ratio is the number that we multiply by to get from the first to the second term. This result would be the same number that gets us from the second to the third term. Likewise, from the third to the fourth and the fourth to the fifth. If this number is not the same, then the sequence is not geometric in nature. If we let 𝑎 denote the first term of the geometric sequence, the second term must be 𝑎 multiplied by the common ratio 𝑟, 𝑎 multiplied by 𝑟. This leads us to the formula for the general term, 𝑎 multiplied by 𝑟 to the power of 𝑛 minus one, when we are trying to work out the 𝑛th term.
In this example, however, we are only going to focus on the first and the second term. The first term 𝑎 is equal to one over 156, and the second term, 𝑎 multiplied by 𝑟, is equal to one over 52. Dividing these two equations gives us 𝑎𝑟 divided by 𝑎 is equal to one over 52 divided by one over 156. Dividing 𝑎𝑟 by 𝑎 gives us 𝑟. One over 52 divided by one over 156 is equal to three. Therefore, the common ratio 𝑟 in this case is equal to three.
We can check that this works for the other terms in our sequence. For example, one over 52 multiplied by three is three over 52. In the same way, three over 52 multiplied by three is nine over 52. And also, nine over 52 multiplied by three is 27 over 52.
As this works for all terms, we can say with assurance that the common ratio 𝑟 is equal to three.