Let A 1 , A 2 , A 3 , … . . be squares such that for each n ⩾ 1 , the length of the side of A n equals the length of diagonal of A n + 1 . If the length of A 1 is 12 cm , then the smallest value of n for which area of A n is less than one, is

Question

Let A 1 , A 2 , A 3 , … . . be squares such that for each n ⩾ 1 , the length of the side of A n equals the length of diagonal of A n + 1 . If the length of A 1 is 12 cm , then the smallest value of n for which area of A n is less than one, is,

MARKS App · Accepted Answer

Let a n be the side length of A n . So, a n = 2 a n + 1 , a 1 = 12 ⇒ a n = 12 × 1 2 n - 1 Now, a n 2 1 ⇒ 144 2 n - 1 1 ⇒ 2 n - 1 > 144 ⇒ n - 1 ≥ 8 ⇒ n ≥ 9

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