the arithmetic average is not less than the geometric average

1/1 __>__ 1

(1+2)/2 __>__ sqr(1x2)

3/2 __>__ 1.414

1.5 __> __ 1.414

(1+2+3)/3 __>__ crt(1x2x3)

2 __>__ cube root of 6

cube both sides to get

8 > 6

assume (1+2 + 3 + .... + n)/n __>__ the nth root of n!

try to show (1+2 + 3 ...+n + n+1)/(n+1) __>__ the n+1th root of (n+1)!

(1+2+...n)/n __>__ nth root of n

add (n+1)/n to both sides

(1+2+...n+n+1)/n __>__ (n!)^(1/n) + (n+1)/n

__>__ (1^(1/n))(2^(1/n))....(n^(1/n)) + (n+1)/n

(n+1)/n is less than (3+1)/3 =1.333...for n greater than 2

show 4/3 < (n+1)^(1/n)

raise both sides to the n power

(4/3)^n < n+1

4/3 < 1+1

16/9 < 3

64/27< 4

assume (4/3)^n < n+1

try to show (4/3)^(n+1) < n+2