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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
(b), (c), (d)
If a be the side of a square, then d=a2
By given condition an=2an+1
or an+1=an2=an1(2)2=an2(2)3=a1(2)n
Replacing n by n-1, we get
an=a1(2)n1=102n1/2
Area of Sn<1a2n<1
1002n1<1 or 200<2n or 2n>200
Now 27=128<200<28=256>200
n=8,9,10(b), (c), (d)

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