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A number is called purple if it can be expressed in the form \(\dfrac{1}{2^a5^b}\) for positive integers a > b. Find the sum of all purple numbers. 

 

I'm not sure how to do this.

 Jul 2, 2020
 #1
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Graphing the possible values for \(a>b\) as positive integers. (Selected Points are the possible values)

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a is y-axis

b is x-axis

 

By plugging in values of A and B...

Notice how if we set b = 1, we can get an infinite series as \(a\) increases by increments of 1.

Notice how if we set b = 2, we can get the same thing except the starting term is different.

 

So the answer really is the sum of an infinite series of infinite series'.... very interesting!

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Let us find the infinite series if b = 1 and \(a\) increases to start off:

 

\(\frac{1}{20}+\frac{1}{40}+\frac{1}{80}...\)

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The formula for an infinite series is

\(\frac{a_{(1)}}{1-r}\)

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So the series for b = 1 is \(\frac{1}{10}\).

 

By applying the same method, we know that the series for b = 2, b = 3, b = 4 is:

1/100

1/1000

1/10000

respectively...

 

So now we have to find the sum to THAT infinite series, which is:

 

\(\frac{\frac{1}{10}}{1-\frac{1}{10}} = \boxed{\text{math is fun}}\)

 

 

 

 

*an interesting thing is as you add the number of different variables, the higher dimensions the graph is. Interesting how we don't need to actually VISUALIZE higher dimensional graphs, we can just calculate the sum if you have the method down.

 Jul 2, 2020
edited by AnExtremelyLongName  Jul 2, 2020

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