Question

# Let S1, S2, … 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
8
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C
9
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D
10
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Solution

## The correct option is D 10We 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=1√2, for all n≥1 Hence, sides of S1, S2,…, Sn form a G.P with common ratio 1√2 and first term 10. ∴ Side of Sn=10(1√2)n−1=102(n−1)/2 ⇒ Area of Sn=(side)2=(102(n−1)/2)2=1002n−1 Now, area of Sn<1 ⇒1002n−1<1 ⇒100<2n−1 ⇒n>7

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