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If a1+a2+a3+......+an=1 AA ai > 0, i=1,...

If `a_1+a_2+a_3+......+a_n=1 AA a_i > 0, i=1,2,3,......,n`, then find the maximum value of `a_1 a_2 a_3 a_4 a_5......a_n`.

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We have
A.M. `ge` H.M
`implies ((S)/(S - a_(1)) + (S)/(S - a_(2)) + ….. + (S)/(S - a_(n)))/(n) ge (n)/((S - a_(1))/(S) + (S - a_(2))/(S) + … + (S - a_(n))/(S))`
or `(S)/(S - a_(1)) + (S)/(S - a_(2)) + …. + (S)/(S - a_(n)) ge (n^(2) S)/(nS - (a_(1) + a_(2) + ...... + a_(n)))`
or `(S)/(S - a_(1)) + (S)/(S - a_(2)) + ... + (S)/(S - a_(n)) ge (n^(2) S)/(nS - S)`
or `(S)/(S - a_(1)) + (S)/(S - a_(2)) +.........+ (S)/(S - a_(n)) ge (n^(2))/(n - 1)`
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