Find the common ratio of a geometric sequence given the middle terms are 56 and 168, respectively.
Remember, the terms of a geometric sequence are found by multiplying the previous term by some common ratio. Alternatively, we can say that some common ratio for the sequence can be found by dividing any term by the term that precedes it. Formally, 𝑟, the common ratio, is 𝑎 sub 𝑛 plus one over 𝑎 sub 𝑛 for values of 𝑛 greater than or equal to one.
Now, we’re told information about the middle terms of the sequence. We don’t know how many terms there are in the sequence. But we can define these consecutively as 𝑎 sub 𝑖 equals 56 and 𝑎 sub 𝑖 plus one equals 168, where 𝑖 is greater than one and less than 𝑛. Then, the common ratio is simply found by dividing the term 168 by the term that precedes it, 56. And 168 divided by 56 is equal to three.
So the common ratio of a geometric sequence whose middle terms are 56 and 168 is three.