Ask Professor Puzzler
Do you have a question you would like to ask Professor Puzzler? Click here to ask your question!
Emmanuel from Papua New Guinea asks, "How do find the common ratio of a geometric sequence if the ratio of the fourth and second term are given?"
Well, Emmanue, the short answer is: you can't!
Let's suppose the second term of geometric is 4, and the fourth term is 16. You might think, "Oh, that's easy - the ratio must be 2, because 4 x 2 is 8, and 8 x 2 = 16!" But that's not necessarily true - maybe the ratio is -2! 4 x -2 = -8, and -8 x -2 = 16.
The problem is, in a geometric sequence, all the even-numbered terms will have the same sign, but that won't tell us anything about the sign of the odd-numbered terms, and that information is needed to find the ratio. But we can set up an equation that'll give us the possible values.
Let's say the second term is 2, and the fourth term is 18. Then
ar = 2, and ar3 = 18
If we rewrite the second equation as ar(r2) = 18 we can subsitute the first equation in place of ar, giving:
2r2 = 18, or
r2 = 9.
Now, it's tempting at this point to say, if r squared is 9, then r must be 3, but you're missing a possibility if you do that, because 9 has two square roots: 3 and -3. These are your two possible ratios. We don't know what the ratio is, but at least we've narrowed it down to two possibilities!
By the way, as a side note, in order to get my students to avoid missing a solution in an equation like r2 = 9, I tell them they have to solve the equation like this:
r2 = 9
r2 - 9 = 0
(r - 3)(r + 3) = 0
Therefore r - 3 = 0 or r + 3 = 0, which leads to r =3 or r = -3.
It's more work, but it keeps them (most of the time) from forgetting a root!