Let $a_i$ , $1\le i\le n$ be non-negative real numbers. Let S denote their sum.Pick out the true statements:

Question

Let $a_i$ , $1\le i\le n$ be non-negative real numbers. Let S denote their sum.Pick out the true statements:

(a)$\prod_{k=1}^{n}{(1+a_k)\ge1+S}$
(b)$\prod_{k=1}^{n}{(1+a_k)\le1+\frac{S}{1!}}+\frac{S^2}{2!}+....\frac{S^n}{n!}$
(c)$\prod_{k=1}^{n}{(1+a_k)\ge \frac{1}{1-S}}$ if $S<1$

Answer 1

(a) and (b) by induction

(c) Take n=1 and have wrong statement. Opposite unequlity is true. Also by induction can be proved.