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If the arithmetic mean of the product of...

If the arithmetic mean of the product of all pairs of positive integers whose sum is n is `A_n` then `lim_(n->oo) n^2/(A_n)` equals to

Text Solution

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`sum_(K=1)^(n-1)k(n-k)`
`sum_(k=1)^(n-1)kn-k^2`
`nsum_(k=1)^(n-1)k-sum_(k=1)^(n-1)k^2`
`n*((n-1)n)/2-((n-1)*n(2n-1))/6`
`(n-1)(n^2/2-(n(2n-1))/6)`
`(n-1)(n^2/6-n^2/3+n/6)`
`(n-1)(n^2/6+n/6)`
`(n-1)(n(n+1))/6`
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