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If Sigma(r=1)^(n) T(r)=n(2n^(2)+9n+13), ...

If `Sigma_(r=1)^(n) T_(r)=n(2n^(2)+9n+13)`, then find the sum `Sigma_(r=1)^(n)sqrt(T_(r))`.

Text Solution

Verified by Experts

`S_(n)=sum_(r=1)^(n)T_(r)=n(2n^(2)+9n+13)`
`rArrT_(r)=S_(r)-S_(r-1)`
`=r(2r^(2)+9r+13)-(r-1)(2(r-1)^(2)+9(r-1)+13)`
=`6r^(2)+12r+6=6(r+1)^(2)`
`rArrsqrt(T_(r))=sqrt6(r+1)`
`rArrsum_(r=1)^(n)sqrt(T_(r))=sqrt6sum_(r=1)^(n)(r+1)`
`sqrt6((n^(2)+3n)/2)`
`=sqrt(3/2)(n^(2)+3n)`
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