If `a_(n)` and `b_(n)` are positive integers and `a_(n)+sqrt2b_(n)=(2+sqrt2))^(n)`, then `lim_(nrarroo) ((a_(n))/(b_(n)))`=

Options

A. `sqrt2`
B. 2
C. `e^(sqrt2)`
D. `e^(2)`

Answer

A

Explanation

We have `a_(n)+sqrt2b_(n)=(2+sqrt2)^(n)`
`rArr" "a_(n)-sqrt2b_(n)=(2-sqrt2)^(n)`
Solving we get `a_(n)=(1)/(2)[(2+sqrt2)^(n)+(2-sqrt2)^(n)]`
and `b_(n)=([(2+sqrt2)^(n)+(2-sqrt2)^(n)])/([(2+sqrt2)^(n)-(2-sqrt2)^(n)])`
`" "=sqrt2([1+((2-sqrt2)/(2+sqrt2))^(n)])/([1-((2-sqrt2)/(2+sqrt2))^(n)])`
Hence, `underset(nrarroo)(lim)((a_(n))/(b_(n)))=sqrt2((1+0)/(1-0))=sqrt2" "(because(2-sqrt2)/(2+sqrt2)lt1)`