1.

If( 1+ x ) ^n= C _ 0+ C _ 1x+C_ 2x ^ 2+ … + C_n x ^ n, thenC _0+2 C_1+ 3C_2+ … + (n + 1 ) C_ nisequal to

Answer»

`2^N+n.2^(n -1) `
`2^(n-1)+ n.2^n`
` 2 ^n + (n+ 1 ) 2 ^n`
` 2 ^(n - 1 )+ (n-1 ) 2 ^n `

Solution : ` ( 1 + x ) ^n = C _0 + C _ 1x +C _ 2x ^ 2+ … +C_n x ^ n `
` RARRX ( 1 + x ) ^n= C _0x+ C _ 1x ^ 2+C _ 2x ^3+… +C_n x ^ ( n+ 1 ) `
Differentiatingonbothsides
`XN (1 +x) ^(n - 1 )+(1 + x ) ^n= C_0+2C_1 x+3 C_2x ^ 2+... ( n + 1 ) C _ n x ^ n `
Substituting`x= 1`inaboveequation,
`thereforen ( 2 )^(n - 1 )+2^n=C_0+2C_1+ 3 C_2+... +(n + 1 )C_ n `


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