InterviewSolution
Saved Bookmarks
InterviewSolution
| 1. |
If `C_(0) , C_(1) , C_(2) ,…, C_(n) ` are coefficients in the binomial expansion of `(1 + x)^(n)` and n is even , then `C_(0)^(2)-C_(1)^(2)+C_(2)^(2)+C_(3)^(2)+...+ (-1)^(n)C_(n)""^(2) ` is equal to . |
|
Answer» Correct Answer - b If n is even, suppose `(r + 1)^(th)` term in the binomial expansion of `(1 + x^(2))` contains `x^(n)`. We have, `T_(r +1) = ""^(n)C_(r) (-1)^(r) (x^(2))^(r) = ""^(n)C_(r) (-1)^(r) x^(2r)` For this term to contain `x^(n)` , we must have `2r = n rArr r = n //2` `therefore ` Coefficients of `x^(n) = ""^(n)C_(n//2) (-1)^(n//2)` Also, Coefficient of `x^(n)` in `(1 - x^(2))^(n)` (see illustration 18) is `C_(0)^(2)-C_(1)^(2)+C_(2)^(2)+C_(3)^(2)+...+ (-1)^(n)C_(n)""^(2)` `therefore C_(0)^(2)-C_(1)^(2)+C_(2)^(2)+C_(3)^(2)+...+ (-1)^(n)C_(n)""^(2)= ""^(n)C_(n//2)(-1)^(n//2)` . |
|