A concrete example, since apparently the internet is incapable of bringing it to the point.
Consider an investment returns \(+50\%\) in the first year and \(-50\%\) in the second year. We start with \(100\).
After year 1: \(100 \cdot 1.5\) = 150
After year 2: \(150 \cdot 0.5\) = 75
Arithmetic mean: \( \frac{1.5+0.5}{2} = 1 \). Falsly claim no gain no loss on average. Wrong. Proof: \( 100 \cdot 1 \cdot 1 \ne 75 \)
Geometric mean \( \sqrt[2]{1.5 \cdot 0.5} \approx 0.866 \). Correct. Proof: \( 100 \cdot 0.866 \cdot 0.866 \approx 75 \)
Reasoning is: Geometric mean takes different base value (of \(150\)) into account.