# Math Help - Airthmetic-geometric means problem?

1. ## Airthmetic-geometric means problem?

i have a question from an analysis paper that i think can be solved using the arithmetic-geometric means inequality. This is the question:

show that if a_1, a_2, a_3, ... , a_n are positive, such that

(a_1)(a_2)(a_3)...(a_n)=1

then

(1+a_1) (1+a_2) (1+a_3) ... (1+a_n) ≥ 2^n

it makes sense because using the AM-GM inequality, 'on average' the a_i are greater than 1, so 'on average you have something greater than
(1+1) (1+1) ...(1+1) = 2^n

but i don't know how to prove it rigourously.
can anyone help?

2. This is true for all x & y: $x^2 + y^2 \ge 2xy$. Therefore in your case, $1 + a_j \ge 2\sqrt {a_j }$.

So it follows that $\prod\limits_{j = 1 }^n {\left( {1 + a_j } \right)} \ge \prod\limits_{j = 1}^n {2\sqrt {a_j } } = 2^n \sqrt {\prod\limits_{j = 1}^n {a_j } }$.