Let `S_(1) , S_(2) ,"……."` be square such that for each `n gt 1 `, the length of a side of `S_(n)` equals the length of a diagonal of `S_(n+1)` . If the length of a side of `S_(1)` is 10cm, then for which of the following values of n is the area of `S_(n)` less than 1 sq cm ?

Explanation

We have, length of a side of
`S_(n)` =length of deagonal of `S_(n+1)`
`implies` Length of a side of `S_(n)=(sqrt2)( " length of a side of " S_(n+1))`
` implies (" Length of a side of " S_(n+1))/(" Length of a side of " S_(n))=(1)/(sqrt2), "for all "nge 1`
` implies " Side of " S_(1),S_(2),S_(3),"....." " from a GP with common ratio "(1)/(sqrt2) " and first term " 10`. ` therefore " Side of " S_(n)=10((1)/(sqrt2))^(n-1)=(10)/((n-1)/(2))`
` implies " Area of " S_(n)=("side")^(2)=(100)/(2^(n-1))`
Now, given area of `S_(n)lt1`
` implies (100)/(2^(n-1))lt1 implies 2^(n-1)gt100gt2^(6)`
` implies 2^(n-1)gt2^(6) implies n-1gt6`
` therefore ngt7 " or " n ge 8`.