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For `a in R |a|gt l`, let `lim(nto oo )((1+3sqrt(2)+.....+3sqrt(n))/(n^(7//3)((1)/((an+1))+(1)/((an+2)^2)+....+(1)/((an+n)^2))))=54` Then the possible values (s) of a is/areA. -6B. 7C. 8D. -9 |
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Answer» Correct Answer - C::D Since, `underset(ntoinfty)(lim)[(1+root(3)(2)+root(3)(2)+ . . .+root(3)(n))/(n^(7//3)((1)/(1n+1)^(2)+(1)/(an+2)^(2)+ . . .+(1)/((an+n)^(2))))]` `=lim_(ntooo)(sum_(r=1)^(n)(r^1//3))/(n^(7//3)sum_(r=1)^(n)(1)/((an+r)^2))=lim_(ntooo)(sum_(r=1)^(n)((r)/(n))^(1//3)(1)/(n))/(sum_(r=1)^(n)(a+(r)/(n))^2(1)/(n))` `=(int_(0)^(1)x^(1//3)dx)/(int_(0)^(1)(dx)/((a+x)^(2)))=54` `implies((3)/(4)[x^(4//3)]_(0)^(1))/([-(1)/(x+a)]_(0)^(1))=54` `implies(3//4)/(-(1)/(a+1)+(1)/(a))=54` `implies(3)/(4xx54)=(1)/(a(a+1))impliesa^(2)+a=72` `implies a^(2)+9a-8a-72=0` `implies a(a+9)-8(a+9)=0` `implies(a-8)(a+9)=0impliesa=8` or -9 Hence, opition c and d are correct. |
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