Saved Bookmarks
| 1. |
If n be a natural number define polynomial f_(n)(x) of n^(th) degree as follows f_(n)(costheta)cosntheta i.e. f_(2)(x)=2x^(2)-1 f_(3)(x)=4x^(3)-3x_(1) Then (x+sqrt(x^(2)-1))^(10)+(x-sqrt(x^(2)-1s))^(10) is equal to |
|
Answer» `f_(10)(x)` `f_((n+1))(x)+f_((n-1))=2x.f_(n)(x)` `f_(n)(x)=1/(2x)[f_(n+1)(x)+f_(n-1)(x)]` Now, Put `x=costheta, impliessqrt(x^(2)-1)=isintheta` `(x+sqrt(x^(2)-1))^(10)+(x-sqrt(x^(2)-1))^(10)=(costheta+isintheta)^(10)+(costheta-isintheta)^(10)` `=2cos(10theta)=2f_(10)(costheta)=2f_(10)(x)` |
|