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Q.

A circle of radius r=4 units is inscribed in an equilateral triangle ABC, then an equilateral triangle is inscribed in the circle, a circle again is inscribed in the later triangle and so on. In this way the process continues infinitely. If r, x1, x2, ..., xn, .... be the radii of these circles respectively, then the sum of radii of all the circles  x1+x2+x3... is equals to

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answer is 4.

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Detailed Solution

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For equilateral  Δr=R2. Hence R = 8
Now inradius r of ABC is the circum radius for
second  Δ. Hence  x1=r2=R4
Similarly
x2=x12=R8,x3=x22=R16,.....,xn=xn12=R2n+1

x1+x2+x3+.....xn+... upto   terms

=R4+R4+R16+....+R2n+1+...   upto   terms

Hence, sum of the radii of all the is R4112=4

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