Let S1 be a square of side a. Another square S2 is formed by joining the mid-points of the sides of S1. The same process is applied to S2 to form yet another square S3, and so on. If A1, A2, A3 .. be the areas and P1, P2, P3,…….. be the perimeters of S1, S2, S3,..., respectively, then the ratio (P1 + P2 +P3+ ......) / (A1 + A2 + A3 +.......) equals :-	Select:
A) 2(1+√2)/a
B) 2(2+√2)/a
C) 2(2+√2)/a
D) 2(1+2√2)/a
E) None of these
Explanation :   side of S1= a, perimeter of S1= 4a, area of S1 = a2   side of S2= a/√2 , perimeter of S2= 2√2a , area of S2 = a2/2   side of S3= a/2 , perimeter of S3= 2a , area of S3= a2/4   So, for P1 + P2 + P3 .... we will have a GP as below   4a + 2√2a + 2a + .....   First term =4a, common ratio = 1/√2, Sum = 4a/( 1-1/√2) = 4√2a/(√2 - 1)   For A1 + A2 + A3....we will have another GP   a2 + a2/2 + a2/4 .....   First term = a2 , common ratio = 1/2 , Sum = 2a2.   Hence, the required ratio :-   i.e (P1 + P2 + P3....) / (A1+ A2+A3....).   =&gt; 4√2a / (√2 - 1) / 2a2.   =&gt; 2√2(√2 + 1)/a.   =&gt; 2(2 +√2)/a .   Answer : 2(2+√2)/a
<result>B</result>
Let S1 be a square of side a. Another square S2 is formed by joining the mid-points of the sides of S1. The same process is applied to S2 to form yet another square S3, and so on. If A1, A2, A3 .. be the areas and P1, P2, P3,…….. be the perimeters of S1, S2, S3,..., respectively, then the ratio (P1 + P2 +P3+ ......) / (A1 + A2 + A3 +.......) equals :-