Explanation:
Given that G.M of a set of n observations is the root of their product.

Geometric Mean (G.M) of a set of n observations is given by:
G.M = (x1 * x2 * x3 * .... * xn)^(1/n)

where x1, x2, x3, ...., xn are the given n observations.

To find the index of the G.M of n observations, we need to simplify the above expression.

Taking the logarithm of both sides, we get:
log(G.M) = (1/n) * log(x1 * x2 * x3 * .... * xn)

Using the law of logarithms, we get:
log(G.M) = (1/n) * (log(x1) + log(x2) + log(x3) + .... + log(xn))

Multiplying both sides by n, we get:
n * log(G.M) = log(x1) + log(x2) + log(x3) + .... + log(xn)

Taking antilogarithm on both sides, we get:
G.M^n = x1 * x2 * x3 * .... * xn

Now, the index of the G.M can be found as follows:

Taking logarithm on both sides, we get:
n * log(G.M) = log(x1) + log(x2) + log(x3) + .... + log(xn)

Dividing both sides by log(G.M), we get:
n = (log(x1) + log(x2) + log(x3) + .... + log(xn)) / log(G.M)

Multiplying both sides by (-1), we get:
n = (-1) * (log(x1) + log(x2) + log(x3) + .... + log(xn)) / (-1) * log(G.M)

n = (-1) * log(x1/x2 * x2/x3 * x3/x4 * .... * xn-1/xn) / (-1) * log(G.M)

n = log(x1/x2 * x2/x3 * x3/x4 * .... * xn-1/xn) / log(G.M)

n = log(x1/x2) + log(x2/x3) + log(x3/x4) + .... + log(xn-1/xn) / log(G.M)

n = (n-1) / log(G.M)

Hence, the index of the G.M is (n-1). Therefore, option (C) is the correct answer.