Analysis - How many numbers in the interval [0,1) contain 5 consecutive 5's?

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SUMMARY

The problem focuses on determining the measure of the set E, which includes all numbers in the interval [0,1) that contain the block "55555" in their decimal expansion. To solve this, one should calculate the measure of the complement of E, which consists of numbers that do not contain the block "55555" at all. The approach involves analyzing various cases based on the number of consecutive 5's present in the decimal expansions, ultimately leading to a solution that leverages concepts similar to those used in Cantor set analysis.

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Homework Statement



Let E be the set of points in [0,1) whose decimal expansion contains somewhere the block 55555. Find the measure of E.


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The Attempt at a Solution



I have a feeling that in order to find the measure of E, I should find the measure of the complement of E and then subtract it from 1. The complement of E would be the points in [0,1) containing no 5's, containing no consecutive 5's, containing 2 consecutive 5's, containing 3 consecutive 5's, containing 3 consecutive 5's, or containing 4 consecutive 5's. I'm not sure how to compute the measure of this set, however.

Any help would be greatly appreciated!
 
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Well, start your program. What the measure of all of the points that contain no 5's at all? This is rather like a Cantor set.
 

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