Let the first and the last numbers be denoted by a and b. Then, their geometric means are denoted by \( \rm m_{i} \).

If there are n geometric means, then the formula for the calculation of the common ratio (r) is

\( \rm r = \left ( \frac{b}{a} \right )^{\frac{1}{n+1}} \).

Given

\( \rm a = 3; b = 243; n = 3\)

\( \rm r = \left ( \frac{243}{3} \right )^{\frac{1}{3 + 1}} \)

\( \rm \therefore r = 3 \)

The nth mean of the series is given by the formula \( m_{n} = a r^{n} \).

The 3rd mean of the series is given by \( m_{3} = 3 \cdot 3^{3} \).

\( \rm \therefore m_{3} = 81 \).

Geometric mean between 3 and 243 is \( \rm GM = \sqrt{3 \cdot 243} = 27 \).

Arithmetic mean between 3 and 243 is \( \rm AM = \frac{3 + 243}{2} = 123 \).

Hence, AM > GM. The difference is \( \rm AM - GM = 123 - 27 = 96 \).