If `a , b , c ,` are positive real numbers, then prove that (2004, 4M) `{(1+a)(1+b)(1+c)}^7>7^7a^4b^4c^4`

Explanation

Using A.M `ge` G.M
`((a_(1))/(2) + (a_(1))/(2) + a_(2) + a_(3) + a_(4) + …. + a_(n))/(n + 1) ge (((a_(1))/(2))^(2).a_(2).a_(3).a_(4)……..a_(n))^((1)/(n + 1))`
`implies ((1)/(n + 1))^(n + 1) ge (a_(1)^(2) a_(2) a_(3) a_(4)….a_(n))/(4)`
`implies (4)/((n + 1)^(n + 1)) ge a_(1)^(2) a_(2) a_(3) a_(4)...... a_(n)`
Hence, required maximum value is `(4)/((n + 1)^(n + 1))`