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Question

# Let S1,S2,S3,…,Sn 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 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 10If a be the side of the square, then d=a√2. By given condition, an=√2an+1 or, an+1=an√2=an−1(√2)2=an−2(√2)3=…=a1(√2)n Replacing n by n−1, we get an=a1(√2)n−1=102(n−1)/2 Area of Sn<1⇒a2n<1 ⇒1002n−1<1 ⇒2n>200 ∴n=8,9,10

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