1.

Ifa _ kisthecoefficientofx ^kintheexpansion of(1 + x+ x ^ 2 ) ^ nfork = 0, 1, 2, …, 2n, thena _ 1+ 2a _ 2 +3a _ 3 +… +2na _ (2n )isequalto

Answer»

` -a _ 0 `
`3^n `
`n. 3^(n + 1 ) `
`n. 3^n `

Solution : ` (1 | x | x ^2 ) ^n=a_0+ a _ 1 x ^ 1+a _ 2 x ^ 2+ a _ 3 x ^ 3+ …. +a _ (an )x ^ (2N )… (1) `
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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