## Ask Professor Puzzler

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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 ar^{3} = 18

If we rewrite the second equation as ar(r^{2}) = 18 we can subsitute the first equation in place of ar, giving:

2r^{2} = 18, or

r^{2} = 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 r^{2} = 9, I tell them they have to solve the equation like this:

r^{2} = 9

r^{2} - 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!