Question

# Let S1, S2, S3, ... 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 then 1cm2

A
7
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B
8
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C
9
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D
10
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Solution

## The correct options are B 8 C 9 D 10The length of a side of Sn= length of diagonal of Sn+1⇒ The length of a side of Sn=√2 (length of side of Sn+1)⇒lengthofthesideofSn+1lengthofthesideofSn=1√2; for n≥1⇒ Sides of S1,S2,⋯,Sn form G.P with common ratio 1√2 and first term is 10∴ Side of Sn=10(1√2)n−1 ⇒ area of Sn=(side)2=1002n−1Now, area of Sn≤1∴1002n−1≤1⇒2n−1≥100⇒n−1≥7orn≥8Hence, options (B), (C) and (D) are correct

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