1. Simply apply geometry , you will find that the radius of circles are R , R/2 , R/4 , .......

 

Hence sum of areas of circles = pi.R² + pi.(R/2)² + pi.(R/4)² + ....... n terms

 

= pi.R²{1 - (1/2)ⁿ}/(1 - 1/2) = 2pi.R²{1 - 1/2ⁿ}

 

2. As n tends to infinity , this area tends to 2pi.R².

 

3. Sides of triangle also decrease in same order as that of radius of circles.

a , a/2 , a/4 , .............

Sum of areas = (3^1/2/2)a²(1 - 1/2ⁿ)

 

4. As n tends to infinity , this area tends to (3^1/2/2)a².

 

 

Mr.IITIAN007 , I would request you to be polite with Moderator. She has been working hard for the fruitful running of the forum.