Question

यदि ` a _ 1 , a _ 2, a _ 3, a _ 4, ( 1 + x ) ^n ` के विस्तार में चार कर्मगत पद है, तब सिद्ध कीजिये कि ` ( a _ 1 ) /( a _1 + a _ 2 ) + ( a _3 ) /( a _ 3 + a _ 4 ) = ( 2a _ 2 ) /( a _ 2 + a _ 3 ) `

Answer 1

माना ` a _ 1 , a _2, a _ 3 ` व ` a _ 4 ` चार कर्मगत पदों r वां पद, ` ( r + 1 ) ` वां , (r + 2 ) वां तथा (r + 3 ) वे पदों के गुणांक है |
तब ` a _ 1 = ""^n C _ ( r - 1 ) , a _ 2 = ""^n C _ r , a _ 3 = ""^ n C _ ( r + 1 ` तथा ` a _ 4 = ""^ n C _ ( r + 2 ) `
अब ` a _ 1 + a _ 2 = ""^n C _ ( r - 1 ) + ""^nC _ ( r ) = ""^ ( n +1 ) C _ ( r ) `
` a _ 2 + a _ 3 = ""^n C _ r = ""^n C _ ( r +1 ) = ""^ ( n + 1 ) C _ ( r + 1 ) `
तथा ` a _ 3 + a _ 4 = ""^ n C _ ( r + 1 ) + ""^n C _ ( r + 2 ) = ""^ (n + 1 ) C_ ( r + 2 ) `
इसलिए ` ( a _ 1 ) /( a _ 1 + a _ 2 ) + ( a _ 3 ) /( a _ 3 + a _ 4 ) = ( ""^C _ ( r - 1 ))/( ""^ ( n + 1 ) C _ r ) + ( ""^ n C _ ( r + 1 ) ) /( ""^ (n + 1 ) C _ ( r + 2 ) `
` = ( n! ) /( ( r - 1 ) ! ( n - r + 1 ) ! ) ( r! ( n + 1 - r )! ) /( (n + 1 ) ! ) + ( n! ) / ( ( r + 1 ) ! ( n - r - 1 ) ! ) * ( ( r + 2 ) ! ( n + 1 - r -2 ) ! ) /( ( n + 1 ) ! ) `
` =( 2 ^n C _ r ) /( ""^ (n + 1 ) C _ ( r + 1 )) = ( 2a _2 ) /( a _ 2 + a _ 3 ) `