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Ifa _ kisthecoefficientofx ^kintheexpansion of(1 + x+ x ^ 2 ) ^ nfork = 0, 1, 2, …, 2n, thena _ 1+ 2a _ 2 +3a _ 3 +… +2na _ (2n )isequalto |
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Answer» ` -a _ 0 ` Differentiating(1)withrespectto .x. `n(1 + x+ x ^ 2 )^(n -1)( 1 + 2x)= a _ 1+2a _ 2x+3a _ 3x ^ 2+ ... +2NA ^(an) x ^( 2n - 1 ) ""`...(2) Put`x= 1` in (2) ` 3N (3) ^(n - 1 )= a _ 1 +2a _ 2+3a _ 3+... +2n a _(2n )` ` thereforea _ 1+2 a _ 2+3a _ 3+ ... +2n a _ (2n)= n.3^(n) ` |
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