# Math Help - Verifying inequalities with induction

1. ## Verifying inequalities with induction

Hey, I've been having trouble with these two inequalities. I did the basis step but I'm having trouble proving for n+1.

1) for n = 1, 2.. and a(i) are positive numbers

$(a_{1}*a_{2}...*a_{2^n})^\frac{1}{2^n} \leq \frac{a_{1} + a_{2} + ... + a_{2^n}}{2^n}$

revised (cartesian product cardinality = cardinality of each set multiplied)
2) $\mid X_{1} \times X_{2} ... \times X_{n} \mid = \mid X_{1} \mid * \mid X_{2} \mid ... \mid X_{n} \mid$

Could you please explain how you arrive to each conclusion? Thanks in advance.

2. $\frac{x_1+x_2+\cdots+x_{2^k}+x_{2^k+1}+\cdots+x_{2 ^{k+1}}}{2^{k+1}} }\\ &=& \frac{( \frac{x_1+x_2+\cdots+x_{2^k}}{2^k} ) + ( \frac{x_{2^k+1}+x_{2^k+2}+\cdots+x_{2^{k+1}}}{2^k} )} {2}$