Brianna L. answered • 07/29/14
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Specializing in Math Anxiety: Always Patient, Never Pushy!
Hi Talo,
Remember that for geometric series, it's a progression by multiplication. From the first term, you multiply by your common ratio to get each successive term. So for this question, if our first term is 2, and our common ratio is r, you would get the fourth term like this: take 2, and multiply it by r a total of three times: 2 * r * r * r. (Why 3? because we're moving up only 3 terms, from the 1st to the 4th. You might have had this explained to you with the expression "n-1": to find the nth term, multiply n-1 times by r, or "rn-1".) We can express 2 * r * r * r more simply, with the expression 2*r3.
So we know that our fourth term is 16. We also know that it can be expressed by 2*r3, so we write 16 = 2*r3. Alright, do some algebra -- we want to get an expression that looks like "r= something". First step is to divide both sides by 2, to try to get that r by itself, and we get 8 = r3. With me so far?
Now all we have to do is take the 3rd root of both sides (also known as the cube root), to get a final "r=" expression. The cube root of r3 is just r; the cube root of 8 is 2 (you can do this on a calculator if you don't know it, but you should be able to recognize that 23=8). So, yay! We know that r=2!
What do we do with this information? Well as I'm sure you've been taught, the GENERAL formula for a geometric sequence is xn = ar(n-1), where xn denotes the nth term, a is the initial value or first term, and r is the common ratio. We can fill in the ones we've found: xn = 2 * 2(n-1). Heck, we can even simplify that to xn = 2n -- this sequence is just the powers of 2!
So now that we have that awesomely simple general formula for our equation, we can find the seventh term. For the 7th term, n=7, so fill that in: x7 = 27 = 128! Woohoo!
If you have questions about any steps I did or anything else, please feel free to ask :)
Brianna L.
