Question

If we have a large circle of radius 'R'. There are two more circles drawn on its diameter of half 'R'.
now the 1st circle touches the two circles on diameter and the large circle also , then we have second circle above 1st circle touching the two circles and the 1st circle then there is a 3rd circle touching 2nd circle an the two cicles on diameter and so on...

Now the radius of the 555 the circle is 1 unit. Find 'R'?

Answer 1

So the radius of 1st circle is R
so the radius of 2nd circle is R/2
 the radius of 3rd circle is R/4

so we see the radius are in GP where r = $\frac{1}{2}$

so using formula for n th term we have R$\frac{1}{2^{n-1}}$ 
so the 555 th circle has radius 1 unit

so $\frac{R}{2^{554}}=1$

so R = $2^{554}$ unit