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We've had several questions recently about how to find terms of a geometric sequence, if you've been given specific information about the sequence, such as, what two of the terms are.
So let's consider a couple examples. For starters, let's say we have a sequence in which the second term is 4, and the fifth term is 27/2. How do we find the first term and the common ratio?
First, since the nth term is arn-1, we know that ar = 4, and ar4 = 27/2. So what do we do with these two pieces of information? Well, we can take the second equation and divide it by the first one, to obtain r3 = 27/8, or r = 3/2.
Since we know that r = 3/2, we can plug that into our first equation, and get (3/2)a = 4, so a = 8/3.
Not too bad, right?
So let's try another one that works out to be just a little different.
If the second term is 24, and the fourth term is 6, find the first term and the common ratio.
We start out the same way:
ar = 24, and ar2 = 6. Dividing these two equations gives r2 = 1/4.
At this point, it's tempting to take the square root of both sides and say r = 1/2. (I see algebra students do this all too often!)
However, it's important to remember that 1/4 has TWO square roots: 1/2 and -1/2.
This means there are TWO possibilities for a. If r = 1/2, then a = 48, but if r = -1/2, then a = -48.
As a general rule of thumb, if you're given two terms of the same parity (both odd numbered terms, or both even numbered terms) you're going to have two possible solutions.
Thanks for asking, and good luck with your sequences!