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Geometric Series - Problem 2

Teacher/Instructor Carl Horowitz
Carl Horowitz

University of Michigan
Runs his own tutoring company

Carl taught upper-level math in several schools and currently runs his own tutoring company. He bets that no one can beat his love for intensive outdoor activities!

Summing a series, so behind me I have a series consisting of three terms and some dot, dot, dot which means that this is ongoing. So what this tells me is that I am summing a infinite series, there is no end, there is no specific sum this many terms. We don't have a formula for summing a infinite arithmetic sequence, so I'm hoping that this is a geometric series that I am dealing with.

So if geometric series we are going from one term to the next by multiplying by a consistent rate.

So what I see is some denominators for our second and third term but not our first, so I'm going to write in the denominator just to make my life a little bit easier to see if I can get any consistencies in this case. So I look at my denominators I'm going from 1 to 4 to 16 which tells me that I have to be multiplying by 4 in the denominator each time to go from one term to the next. So I know my rate is going to be something over 4.

Similarly I'm going from 5 to -15 to 45 in the numerator which tells me I'm multiplying by 3, but there is this sign change, I'm going from positive to negative and back to positive, the only way to do that is if we have a negative involved in there as well, so I know that my rate then has to be -3 over 4.

You can always check 5 times -3/4 is -15 over 4 times -3/4 is 45 over 16, so I found the rate for this geometric infinite series.

We now need to find the sum. We have a formula for the infinite sum; s is equal to a1 over 1 minus r, now all we have to do is plug in our information. We know that a1 is 5, that's easy enough and we know that our rate is -3/4 so this just becomes 1 minus -3/4. 1 minus -3/4 just becomes positive so this just becomes plus, so this becomes 5 over 1 plus 3/4, four-fourths plus 3/4 is just 7/4 dividing by a fraction just flip and multiply, so this is 5 times 4 over 7 which just leaves us with 20 over 7.

So finding a infinite sum, make sure it's a geometric series because in order for it to have an infinite sum, it has to be geometric, find your rate and then just plug it into your equation.

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