if $a_1,a_2,a_3,\ldots,a_n≥0$ then the arithmetic mean is:
$A_n=\frac{a_1+a_2+a_3+\cdots+a_n}{n}$
and the geometric mean is:
$G_n=\sqrt[n]{a_1a_2a_3\ldots a_n}$
(a) Making use of the fact that $G_n≤A_n$ when $n=2$ prove by induction over $K$ that $G_n≤A_n$ for all $n=2^k$
(b)For a general n with $2^m<n$ .Aply the part (a) to the $2^m$ numbers.
$a_1,\ldots,a_n,\underbrace{A_n,\ldots,A_n}_{2^m-n\text{ times}}$