Q13/076) prove that (1+x) * ( 1 + x^2) * (1+x^3) * (1+ x^4) … (1+x^n) >= (1+x^(n+1)/2)^n

we have x^a + x^b >= 2x^(a+b)/2 by AM GM inequality

adding 1 + x^(a+b) on both sides we get

(1+x^a)(1+x^b) >= 1 +  2x^(a+b)/2 + x^(a+b) = ( 1+ x^(a+b)/2)^2

Putting b = n+ 1 -a we get

(1+x^a)(1+x^(n+1- a) >= (1+ x^(n+1)/2)^2

For n even taking a from 1 to n/2 we get n/2 expressions and multiplying them out we get

(1+x)(1+x^2)( 1 + x^3) .. (1+x^n) >= (1+ x^(n+1)/2)^n as

 For n odd we have n-1 ( running a from 1 to (n-1)/2 we get (n-1)/2 pairs and

As (1+x^(n+1)/2)= (1+x^(n+1)/2) we get 1 term and multiplying we get the result