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Question

Let S1, S2, 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
8
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C
9
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D
10
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Solution

The correct option is D 10
We have, length of a side of Sn is equal to the length of a diagonal of Sn+1.
Hence,
Length of a side of Sn=2(Length of a side of Sn+1)
Length of a side of Sn+1Length of side of Sn=12, for all n1
Hence, sides of S1, S2,, Sn form a G.P with common ratio 12 and first term 10.

Side of Sn=10(12)n1=102(n1)/2
Area of Sn=(side)2=(102(n1)/2)2=1002n1
Now, area of Sn<1
1002n1<1
100<2n1
n>7

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