Question

Let S₁, S₂,… be squares such that for each n ³ 1, the length of a side of S_n equals the length of a diagonal of S_n+1. If the length of a side of S₁ is 10 cm, then the smallest value of n for which Area (S_n) < 1 is –

• A. 7
• B. 8
• C. 9
• D. 10

Answer 1

Answer: (B)

8

Answer 2

Let a_n denote the length of a side of S_n. We are given

Length of a side of S_n = Length of a diagonal of S_n+1

̃ a_n = $\frac { 1 } { \sqrt { 2 } }$.

Thus, a₁, a₂, a₃,.....is a G.P. with first term 10 and common ratio 1/$\sqrt { 2 }$.

Therefore, a_n = 10(1/$\sqrt { 2 }$)^n–1

Also, Area (S_n) = a < 1

̃ [10 (1/$\sqrt { 2 }$)^n–1]² < 1

̃ 100 < 2^n–1

̃ Smallest possible value of n is 8.