InterviewSolution
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InterviewSolution
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If `C_(0),C_(1), C_(2),...,C_(n)` denote the cefficients in the expansion of `(1 + x)^(n)`, then `C_(0) + 3 .C_(1) + 5 . C_(2)+ ...+ (2n + 1) C_(n) = ` .A. `n.2^(n)`B. `(n-1)2^(n)`C. `(n+1)2^(n+1)`D. `( n+1)2^(n)` |
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Answer» Correct Answer - d We have, `c_(0) + 3 C_(1) + 5 C_(2) + ...+(2n +1)C_(n)` `sum_(r=1)^(n) (2r +1) C_(r)` `sum_(r=1)^(n) (2r +1)""^(n)C_(r)" "[because C_(r) = ""^(n)C_(r)`] `sum_(r=1)^(n) (2r. ""^(n)C_(r)+""^(n)C_(r))` `sum_(r=1)^(n) 2r. ""^(n)C_(r)+sum_(r=1)^(n)""^(n)C_(r)` `2sum_(r=1)^(n) r. ""^(n)C_(r)+sum_(r=1)^(n)""^(n)C_(r)` `2sum_(r=1)^(n) r.(n)/(r) ""^(n-1)C_(r-1)+sum_(r=1)^(n)""^(n)C_(r)" "[because ""^(n)C_(r)= (n)/(r) .""^(n-1)C_(r-1)]` `2nsum_(r=1)^(n) ""^(n-1)C_(r-1)+sum_(r=1)^(n)""^(n)C_(r)` `2nsum_(r=1)^(n) 2^(n-1)+2^(n) = n.2^(n) + 2^(n) = (n-1)2^(n)` . |
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