The Geometric Mean (G.M) makes index numbers time-reversible. This can be explained as follows:

Index numbers are used to measure changes in the level of a certain phenomenon over time. For example, the Consumer Price Index (CPI) measures the changes in the prices of goods and services over time. The index number is calculated by taking the ratio of the current value of the phenomenon to its value in a base period and multiplying it by 100.

One of the important properties of an index number is its time-reversibility. This means that if we reverse the time series, the resulting index number should be the reciprocal of the original index number. For example, if the index number for year 2 relative to year 1 is 120, then the index number for year 1 relative to year 2 should be 1/1.2 = 0.8333.

The Geometric Mean formula for calculating index numbers has the property of time-reversibility. The formula for the index number is:

Index Number = (Product of (Current Value/Base Value)^(1/n)) x 100

where n is the number of periods.

The Geometric Mean formula uses the product of the ratios of current value to base value raised to the power of 1/n. This formula has the property that if we reverse the time series, the resulting index number is the reciprocal of the original index number. Therefore, the Geometric Mean formula is said to be time-reversible.

In contrast, the Arithmetic Mean (A.M) and Harmonic Mean (H.M) formulas for calculating index numbers do not have the property of time-reversibility. Therefore, they cannot be used to calculate time-reversible index numbers.

Hence, the Geometric Mean formula is the preferred method for calculating index numbers as it has the property of time-reversibility.