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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 10cm, then for which of the following values of n is the area of Sn less than 1sq.cm?

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 10If a be the side of the square, then diagonal d=a√2.By hypothesis⇒an+1=an√2=an−1(√2)2=an−2(√2)3=...=a1(√2)n=∴an+1=a1(√2)n⇒an=a1(√2)n−1=102(n−1)2∵ Area of Sn<1⇒a2n<1⇒1002n−1<1⇒2n>200>27⇒2n>27⇒n>7 (Since bases are same we can equate the powers)∴n=8,9,10

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