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If(1+ x ) ^(15)= a _ 0+ a _1 x+ … + a _(15) , thensum _(r = 1 ) ^(15)r(a _ r )/(a _ (r-1))is equalto |
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Answer» 110 `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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