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Geometric Sequences  Problem 1
Carl Horowitz
Carl Horowitz
University of Michigan
Runs his own tutoring company
Carl taught upperlevel math in several schools and currently runs his own tutoring company. He bets that no one can beat his love for intensive outdoor activities!
Using the general term of a geometric sequence to find a specific term number. So what I have behind me is a geometric sequence and I know this because I'm looking at my terms and in order to get from one term to the next, I always I'm doing the same exact thing dividing by 3.
So to go from 27 to 9, I know I have to divide by 3, in order to make the 9 negative, I know I need a negative in there. Same thing if I'm going from 9 to 3, divide by 3 in order to turn it from a negative to a positive once again I know there has to be a negative.
So I know that my r, my rate is going to be, oops, r is always what we multiply by, so if I said r was 3, we'd actually be multiplying by 3 and our numbers would be going up, but we're multiplying by something that makes our terms smaller, so I actually know that my r has to be a fraction, I'm dividing by 3 which means I'm actually multiplying by 1/3.
So my rate is what I'm multiplying by 1/3 in this case. My first term is just 27, so I know that a sub n is equal to a1 times r to the n minus 1, all I have to do is plug things in. A sub n is equals to a1, 2 is 27 times 1/3 to the n minus 1. So there is my general term, the first thing we're asked for.
The second thing we're asked for is a sub 8. For that all we have to do is plug in n equals 8, a sub 8 is equals to 27 times 1/3 to the 8 minus 1 or to the 7th.
A lot of times with a geometric sequences what's going to happen is these terms are going to be massive, this negative a little bit clear, so often times at least in my classes I'm okay leaving it like this. In this particular example though we can simplify it up without too much work and how we can do that is 27 is the same thing as 3 to the third. So what I really have here is 3 to the third times, when we have a fraction to a power, the power goes to both things, 1 to the 7th is just 1 and 3 to the 7th I don't know what it is, but I can just leave it as 3 to the 7th. Dealing with a fraction when we are dividing and our bases are the same, it just turns into subtraction so 3 to the third to 3 to the 7th is 3 to the 1/4, the negative just puts it on the bottom, so what we end up with is 1 over 3 to the fourth to 1 over 81.
Often times these numbers are going to be too hard to do that, but for this particular example, using laws of exponents, I can simplify this down a little more.
So taking a sequence, identifying as geometric and then finding our general term and in turns another term in the sequence.
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Carl Horowitz
B.S. in Mathematics University of Michigan
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