Question 889285
<pre>
Here is a more advanced way of finding them, using the
Cauchy AM-GM inequality.

Both the arithmetic and the geometric means of any three 
unequal numbers is always between the smallest and largest.

The geometric mean of three numbers is the cube root of
their product.

Therefore the geometric mean of the three consecutive integers,
by calculator is approximately 4.932424149

By the Cauchy inequality, the geometric mean of any number of 
unequal positive numbers is always less than their arithmetic 
mean.  

The arithmetic mean of three positive consecutive integers 
is always the middle one. 

The only three consecutive integers which 4.932424149 could 
be between the smallest and largest of and less than the 
middle one, are the positive integers 4,5, and 6.

Their sum is 4+5+6 = 15

Check: 4󬊆 = 120.

Edwin</pre>