InterviewSolution
Saved Bookmarks
InterviewSolution
| 1. |
If `C_(0), C_(1), C_(2),..., C_(n)` denote the binomial coefficients in the expansion of `(1 + x)^(n)` , then . `1^(2). C_(1) - 2^(2) . C_(2)+ 3^(2). C_(3) -4^(2)C_(4) + ...+ (-1).""^(n-2)n^(2)C_(n)=`. |
|
Answer» Correct Answer - a We have, `1^(2) . C_(1) - 2^(2) . C_(2) + 3^(2) .C_(3) - 4 ^(2) .C_(4) + ...+ (-1)^(n-1) n^(2).C_(n)` `=sum_(r=1)^(n) (-1)^(r-1) r^(2) ""^(n)C_(r) [because ""^(n)C_(r) = C_(r)] ` ` = sum _(r=1)^(n) (-1)^(r-1) [ r (r -1) +r] ""^(2)C_(r)` ` = sum _(r=1)^(n) (-1)^(r-1) r(r-1)""^(n)C_(r) + sum_(r=1)^(n) r (-1)^(r -1)""^(n)C_(r)` `sum _(r=1) ^(n) (-1)^(r -1) r (r -1)(n)/(r). (n-1)/(r-1) ""^(n-2)C_(r - 2) + sum _(r=1)^)(n) r(-1)^(r-1) (n)/(r)n ""^(n-1)C_(r -1)` ` n(n-1)sum_(r=1)^(n) (-1)^(r-1) ""^(n-2) C_(r-2) + n sum_(r=1)^(n) (-1)^(r-1) ""^(n-1)C_(r-1)` `= n(n -1){sum_(r=2)^(n) (-1)^(r -1)""^(n-2)C_(r-2)}+n {sum _(r=1)^(n) (-1)^(r-1) ""^(n-1)C_(r-1)}` `= n(n-1)xx0 + n xx - = 0` . |
|