Karylle from Marinduque wants to know how you can find the ratio in an infinite geometric series, if you know that the sum is a particular multiple of the first term.

This is in that interesting class of problems in which you feel like you don't have enough information to solve it. After all, there are a lot of unknowns (the first term, the ratio, and the sum) and you'll only have one equation to work with. Whenever my students have problems like this, I tell them to simplify the equation as much as possible, and see what happens!

So let's take an example.  Let's say that you know the sum of the infinite series is 5 times the first term. Can you find the common ratio?

Well, the sum of the series is a/(1 - r), and that is 5a.

a/(1 - r) - 5a = 0

Ah...I see what's going to happen already - the a is going to factor out:

a(1/(1 - r) - 5) = 0

so either a = 0 (in which case, r could be anything, right? It's a really boring series, with all the terms equal to zero, but hey, it works!) or 1/(1 - r) = 5. This leads to:

5 - 5r = 1

5r = 4

r = 4/5

I would be inclined to say that the problem should be reworded to state that the first term is not 0, in order to avoid having "r can be any real number" as the answer. If you add in that proviso, then the answer is r = 4/5.

Hope that helps, Karylle!

Professor Puzzler

Jana from the Philippines wants to know, if you know the second and fourth terms, how do you find the first term?

Well Jana, that's a bit of a trick question, because if you don't tell me what KIND of sequence it is, I can't figure out the first term (or any other terms for that matter!). The most common types of sequences you might be talking about are: arithmetic and geometric. Of course, that's not all the possiblities. It could be a Fibonacci sequence, for example, or even a random list of numbers. Of course, if the sequence is random, there's really no way we can figure out any terms, is there?

So let's suppose the sequence is either arithmetic or geometric. And let's use the same numbers for both types of sequences.  We'll say that the first term is 12, and the fourth term is 48.

Arithmetic

The nth term of an arithmetic sequence is given by a+(n-1)d, where a is the first term, and d is the common difference.

Thus, 12 = a + d, and 48 = a + 3d.  This is a system of two equations in two unknowns, and if we solve it, we find that d = 18, and therefore, a = -6. That's our first term.

Geometric

Then nth term of a geometric sequence is given by ar^(n -1). Thus, 12 = ar, and 48 = ar^3.  If we divide the second equation by the first one (which we can only do if a and r are not zero!) we obtain 4  = r^2, from which r is either 2 or -2.  Thus, the first term is 6 or -6.

Just for Kicks

A Fibonacci sequence is a sequence in which each term is the sum of the two previous terms. So if this was a Fibonacci sequence, the third term would have to be 36, since 12 + 36 = 48.  And this leads to the first term being 24, since 24 + 12 = 36.

There you go! How to find the first term, given the 2nd and 4th terms, for three types of sequences. 

Good luck!
Professor Puzzler