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

Options

A. 7
B. 8
C. 9
D. 10

Answer

B, C, D

Explanation

We have, length of a side of `S_(n)` is equal to the length of a diagonal of `S_(n+1)`. Hence,
Length of a side of `S_(n)=sqrt2` (Length of a side of `S_(n+1)`)
or `("Length of a side of" S_(n+1))/("Length of side" S_(n))=1/sqrt2`, for all `nge1`
Hence, sides of `S_(1),S_(2),..,S_(n)` form a G.P. with common ratio `1//sqrt2` and first term 10.
`therefore` Side of `S_(n)=10(1/(sqrt2))^(n-1)=10/(2^((n-1)/2))`
`rArr` Area of `S_(n)=("side")^(2)=(10/(2^((n-1)/2)))^(2)=100/(2^(n-1))`
Now, area of `S_(n)lt1`
`rArr2^(n-1)gt100`
`rArrngt8`