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Let S1, S2, be squares such that for ea...

Let `S_1, S_2, ` be squares such that for each `ngeq1,` the length of a side of `S_n` equals the length of a diagonal of `S_(n+1)dot` If the length of a side of `S_1i s10c m ,` then for which of the following value of `n` is the area of `S_n` less than 1 sq. cm? a. 5 b. 7 c. 9 d. 10

A

7

B

8

C

9

D

10

Text Solution

Verified by Experts

The correct Answer is:
B, C, D
Let `a_(n)` denotes the length of side of the square `S_(n)`.
We are given `a_(n)` = length of diagonal of `S_(n +1)`.
`rArr a_(n) = sqrt2 a_(n+1)`
`rArr a_(n +1) = (a_(n))/(sqrt2)`
This shows that `a_(1), a_(2), a_(3)`,.. Form a GP with common ratio `1//sqrt`.
Therefore, `a_(n) = a_(1) ((1)/(srt2))^(n-1)`
`rArr a_(n) = 10 ((1)/(sqrt2))^(n-1) " " [ :' a_(1) = 10," given"]`
`rArr a_(n)^(2) = 100 ((1)/(sqrt2))^(2(n-1))`
`rArr (100)/(2^(n-1)) le 1 " " [ :' a_(n)^(2) le 1" given"]`
`rArr 100 le 2^(n -1)`
This is possible for `n ge 8`
Hence, (b), (c), (d) are the correct answer
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