Let `S_(1),S_(2),"…."` be squares such that for each `n ge 1`, the length of a side of `S_(n)` equals the lengh of a diagonal of `S_(n+1)`. If the length of a side of `S_(1)` is 10 cm and the area of `S_(n)` less than 1 sq cm. Then, find the value of n.

Options

A. 7
B. 8
C. 5
D. 6

Answer

B

Explanation

We have,
Length of a side of `S_(n)` = Length of a diagonal of `S_(n+1)`
`rArr" "` Length of a side of `S_(n)=sqrt(2)` Length of a side of `S_(n+1)`
`rArr" "("Length of a side of "S_(n+1))/("Length of a side of "S_(n))(1)/(sqrt(2))"for all "ngtl`.
`rArr" "` Sides of `S_(1),S_(2) , . . . .S_(n)` form a G.P. with common ratio `(1)/(sqrt(2))` and first term 10.
`:.` Length of the side of `S_(n)=10((1)/(sqrt(2)))^(n-1)=(10)/((n-1)/(2^(2)))`
`rArr" Area of "S_(n)=("side")^(2)=((10)/((n-1)/(2^(2))))^(2)=(100)/(2^(n-1))`
Now,
Area of `S_(n)lt1`
`rArr" "(100)/(2^(n-1))lt1rArrn-1ge7rArrnge8`