106Q1. (1+2+3+...+n)n > (1.2.3...n)1/n (AM-GM)
==> [(n+1)2]n > n!
Proved
Q.1) prove : (n+1 / 2)n ≥ n!
Q.2) prove : 2n + x Cn .2n-x Cn ≤ (2nCn)2
Q.3) If s be the sum of n +ve unequal quantities a,b,c,..... then prove that :
s/s-a + s/s-b + s/s-c +...... > n2 / n2-1
-
UP 0 DOWN 0 0 3
3 Answers
106106Q2. 2n+xCn2n-xCn
= (2n+x)!(2n-x)!n!n!(n-x)!(n+x)!
= (n+1+x)(n+2+x)...(2n+x)(n+1-x)(n+2-x)...(2n-x)n!n!
= [(n+1)2-x2][(n+2)2-x2][(n+3)2-x2]...[4n2-x2]n!n!
is maximum when x=0
So max(2n+xCn2n-xCn) = (2nCn)2
62Q3 proved here: http://www.targetiit.com/iit-jee-forum/posts/a-p-g-p-10650.html
