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The value of limn(n(n+1)2n+1+n(n+2)2(2n+2)+n(n+3)3(2n+3).....+12n3) is

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Solution

nr=1n(n+r)r(2n+r)

nr=1nn2(1+rn)rn(2+rn)

1nnr=11(1+rn)rn(2+rn)

10dx(1+x)x(2+x)=10dx(1+x)x2+2x

=10dx(x+1)(x+1)21=21dttt21

=[sec1t]21=sec1(2)sec1(1)

=π30=π3
Which is the required answer.

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