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Q.

Let S1, S2, ... be squares such that for each n ≥ 1, the length of a side of Sn equals the length of a diagonal of Sn+1. If the length of a side of S1 is 10 cm, then for which of the following values of n is the area of Sn less than 1 square cm?

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a

7

b

8

c

9

d

10

answer is B.

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Detailed Solution

Let a be the side of the square, then diagonal d=a2Given, an=2an+1⇒an+1=an2=an-122=an-223=....=a12n∴an+1=a12n⇒an=a12n-1=102n-12Now, area of sn<1⇒an2<1⇒1002n-1<1⇒2n>200>27⇒n>7∴ n = 8, 9, 10.

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Let S1, S2, ... be squares such that for each n ≥ 1, the length of a side of Sn equals the length of a diagonal of Sn+1. If the length of a side of S1 is 10 cm, then for which of the following values of n is the area of Sn less than 1 square cm?