ShmoopTube

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Infinite Geometric Series, a la Shmoop. Pepe holds the world record as the fastest [Pepe with his trophy in front of lots of pizza boxes]

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pizza eating human.

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Back in 2010 he inhaled 10 slices in 60 seconds.

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He was crowned the “Pizza King” and has held the record ever since.

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But in the years since, he has let himself go.

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He can barely muster up the motivation to make his way to the phone to order one.

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Still, he finds a way.

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One day he is sitting in a pizza joint with some friends when he is challenged to a pizza [Pepe sat with his friends and a pizza is chucked onto the table]

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eating contest by an unknown stranger.

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Pepe wants to relive the glory days, but he isn’t sure he has what it takes.

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Just to be safe, he first eats half a pizza.

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Then one-sixth, one-eighteenth, one-fifty-fourth, and so on...

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If he keeps eating the pizza this way forever, how much pizza will he have eaten in total?

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Well, first let's look at the numbers we have. [Boy looking through binoculars]

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One-half, one- sixth, one-eighteenth, one-fifty-fourth...and so on...

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…looks like we have something called a series.

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Or actually, since we're assuming he'll eat forever... an infinite series, [Pizza slowly disappearing and a clock ticking]

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But let's take a closer look at the numbers and we notice a pattern.

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Between the first two terms, one-half and one-sixth, there's a ratio of one-third.

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Between one-sixth and one-eighteenth, we have the same ratio of one-third.

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Between one-eighteenth and one-fiftieth fourth, the same ratio, one-third.

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We can call this number one-third, the common ratio, or r, of the series. [Old women answers the phone]

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Because the terms of the series are separated by a constant ratio, we can describe the series

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even more specifically, as an infinite geometric series.

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Now that we've identified the type of numbers we're dealing with and the common ratio between [Fraction locked behind bars]

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them, let's get back to the problem.

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If we want to find the total amount of pizza Pepe ate, we should add all the slices together

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and find the sum.

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But wait, if he's eating infinite slices, how can we find a finite sum?

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No worries, Pepe… the sum of an infinite geometric series has a finite sum as long

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as the absolute value of the common ratio is less than one.

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In this case, the absolute value of one-third is one-third, which is less than one. [Teacher at the front of class writing on a whiteboard]

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As you can see from the series above, the numbers are getting smaller and smaller…

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…one-fifty-fourths would go to one over 162, which is getting closer and closer to 0.

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So as you add smaller and smaller numbers, the addition of such small numbers doesn't

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matter much.

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Ok, so let's find that total for Pepe. [Addition of fractions formula]

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The formula for the sum of an infinite geometric series equals the first term divided by the

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quantity one minus the common ratio.

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Substituting our pizza values into the formula, we see that the sum will equal one half divided

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by the quantity one minus one third.

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One minus one-third equals two-thirds, so we're left with one half divided by two thirds.

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Instead of dividing one half by that nasty fraction two thirds, let’s multiply by its

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reciprocal three over two, to get one half times three halves, or 3-fourths.

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That’ll be three quarters of a pizza.

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You can do that, right Pepe? [Pepe picks up the 3/4 of pizza]

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We have faith in you. Here's the water, and the Pepto-bismol and the barf bag... [Arms hold out water etc.. for Pepe]

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