1.

If(1+ x ) ^(15)= a _ 0+ a _1 x+ … + a _(15) , thensum _(r = 1 ) ^(15)r(a _ r )/(a _ (r-1))is equalto

Answer»

110
115
120
135

Solution : ` ( 1 + X ) ^(15)= a _ 0+a _ 1 x +… + a _(15 ) x ^(15) `
`thereforea _ 0= ""^(15) C_0, a _ 1= ""^(15) C_1a _ 2= ""^(15) C_2, … `
` a_(15)= ""^(15) C_(15) `
`thereforer (a_r)/( a_(R-1 ))= r. (""^(15) C_r ) /(""^(15)C _ (r - 1 )) `
`= r (15 - r + 1 )/(r ) "" [ because(""^N C _ r)/(""^nC_(r- 1)) = (n - r + 1 )/(r) ] `
` =15 -r + 1`
`thereforesum_(r = 1 ) ^(15)r . (a_r)/(a _ (r- 1 ))=sum _ ( r = 1 ) ^(15 )r(""^(15) C_r ) /(""^(15) C_(r-1)) `
`=15+14+13 +... +1`
`=(15 (16))/(2) `
` [ because1 +2+...+n =( n ( n + 1 ) ) /(2) ] `
` = 120`


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