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in Algebra by (53.3k points)

Read the following passage and answer the questions:

Let ABCD be 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 An the area of the nth square

(i) The value of α for which \(\sum\limits_{n=1}^\infty A_n =\frac 83\) 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 nth 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 Pn denotes the perimeter of the nth square then \(\sum\limits_{n=1}^\infty\) Pn equals

(a) 8/3

(b) 32/3

(c) 16/3

(d) \(\frac83(4 + \sqrt{10})\)

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1 Answer

+1 vote
by (53.5k points)

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

(ii) (c) 1/2

(iii) (d) \(\frac83(4 + \sqrt{10})\)

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