(i) The value of α for which Σ An = 8/3 ; n=1,∞ is (a) 1/3, 2/3 (b) 1/4, 3/4 (c) 1/5, 4/5 (d) 1/2 ​

Question

Read the following passage and answer the questions :- 

Let ABCD is a unit square and 0 < α < 1. Each side of the square is divided in the ratio α : 1–α, as shown in the figure. These points are connected to obtain another square. The sides of new square are divided in the ratio α : 1–α and points are joined to obtain another square. The process is continued indefinitely. Let an denote the length of side and A_n the area of the nth square

(i) The value of α for which \(\sum\limits_{n=1}^\infty\) A_n = 8/3 is

(a) 1/3, 2/3

(b) 1/4, 3/4

(c) 1/5, 4/5

(d) 1/2

(ii) The value of α for which side of n^th square equals the diagonals of (n + 1)^th square is

(a) 1/3

(b) 1/4

(c) 1/2

(d) 1/ 2

(iii) If a = 1/4 and P_n denotes the perimeter of the n^th square then \(\sum\limits_{n=1}^\infty\) P_n equals

(a) 8/3

(b) 32/3

(c) 16/3

(d) \(\frac 83 (4 + \sqrt{10})\)

Answer 1

(i) (b) 1/4, 3/4

(ii) (c) 1/2

(iii) (d) \(\frac 83 (4 + \sqrt{10})\)