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What is the total area in blue knowing that the area of the large triangle is 1?

 

 May 19, 2020
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I believe that the first (largest) blue areas of the triangle form 3/4ths of the complete triangle.

The next blue area forms 3/4ths of 1/4th of 1/4th of the triangle.

The next blue area forms 3/4ths of 1/4th of 1/4th of 1/4th of 1/4th of the triangle.

Etc.

 

(3/4)  +  (3/4)(1/4)2  +  (3/4)(1/4)4  +  (3/4)(1/4)6  +  ...

 

This forms a geometric sequence with  first term  =  3/4  and with  common ratio  =  1/16:

Sum  =  (3/4) / [ 1 - 1/16 ]  =  (3/4) / (15/16)  =  4/5.

 May 19, 2020

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