Think about it this way. The arithmetic mean of a bunch of numbers is the number a that satisfies
x₁ + x₂ + x₃ + .... = a + a + a + ... + a
The geometric mean is the number b that satisfies
x₁ * x₂ * x₃ * ... = b * b * b * ... * b
There is also a harmonic mean, which is the number h that satisfies
1/x₁ + 1/x₂ + 1/x₃ + ... = 1/h + 1/h + ... + 1/h.