What will be the geometric mean of the following numbers ?
2, 0, 3, 6

Question

What will be the geometric mean of the following numbers ? 
2, 0, 3, 6

Accepted Answer

The correct option is 2:

Students, let's look closely at the concept of the Geometric Mean (GM). The Geometric Mean is primarily used for finding the average rate of change or growth rates and is calculated by finding the $n$-th root of the product of $n$ observations. The fundamental formula is $GM = \sqrt[n]{x_1 	imes x_2 	imes x_3 	imes \ldots 	imes x_n}$. In our dataset (2, 0, 3, 6), one of the values is 0. If any value in the dataset is zero, the product of all the numbers will automatically be zero (2 × 0 × 3 × 6 = 0). Consequently, the $n$-th root of zero, or the geometric mean, will also be zero. Therefore, the geometric mean of the set is 0, making Option 2 the correct choice. It is a critical mathematical rule that the geometric mean becomes zero whenever even one observation in the series is zero.

The incorrect option is 1 (Value: 1): The geometric mean cannot be 1 because the product of the numbers is 0. The only way the GM could be 1 is if the product of the numbers equaled 1.

The incorrect option is 3 (Value: 2): The geometric mean is not 2. To calculate 2 as the GM, the product of the four numbers would need to be $2^4 = 16$. However, since the product is 0, the GM must be 0.

The incorrect option is 4 (Value: 3): The geometric mean is not 3. If the GM were 3, the product of the numbers would have to be $3^4 = 81$. Since the product of the given series is 0, 3 is an incorrect result.