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Question

Let S1,S2,S3,,Sn be squares such that for each n1, 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 sq. cm?

A
7
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B
9
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C
8
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D
10
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Solution

The correct option is D 10
If a be the side of the square, then d=a2.
By given condition, an=2an+1
or, an+1=an2=an1(2)2=an2(2)3==a1(2)n
Replacing n by n1, we get
an=a1(2)n1=102(n1)/2
Area of Sn<1a2n<1
1002n1<1
2n>200
n=8,9,10

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