1.

If|x| lt (1)/(2),thenthe coefficientofx^rintheexpansionof(1 + 2x ) /((1 - 2x)^2), is

Answer»

` r2 ^r `
` (2R - 1 ) 2 ^r `
` r2^( 2r + 1 ) `
` (2r + 1 ) 2^r `

Solution : `( 1 +2X ) /((1- 2x ) ^ 2 ) `
`( 1 + 2x )(1 -2x) ^ ( -2 ) `
`=(1+2x ) [ 1 +(2 ) /(1! )2x+((2) (3))/(2!)(2x) ^ 2+((2)(3) …(2 +r - 2 ))/((r - 1 ) !)( 2x ) ^( r - 1 )+((2)(3) …(2 + r- 1 )) /(r !)( 2x)^ r+...] `
`therefore` Co- efficientof X. is
` 1.[ ((2)(3)... (r + 1 ))/(r! )2 ^ r] +2 [ ((2) (3)... (r ))/((r - 1 ) !)] 2^(r - 1 )`
`= (( r+ 1 ) ! ) /(r ! )2 ^r+(r! ) /((r - 1 ) ! ) 2 ^ r =(r+ 1 +r ) 2 ^ r `
`= (2r + 1 ) 2 ^r`


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