To find the sum of the first three terms of a geometric progression (G.P.) where the first term \( a \) and the fourth term \( ar^3 \) are given, we set up the following equations:

Given:
- \( a = 6 \)
- \( ar^3 = 162 \)

 Find the common ratio \( r \)

From the equation for the fourth term:

\(ar^3\) = 162

Substituting \( a \):

\(6r^3\) = 162

Dividing both sides by 6:

\(r^3 = \frac{162}{6}\) = 27
 

Taking the cube root:

r = \(\sqrt[3]{27}\) = 3

 Find the first three terms

The first three terms of the G.P. are:
- First term: \( a = 6 \)
- Second term: \( ar = 6 \cdot 3 = 18 \)
- Third term: \( ar^2 = 6 \cdot 3^2 = 6 \cdot 9 = 54 \)

Calculate the sum of the first three terms

The sum \( S \) of the first three terms is:

S = a + ar + ar\(^2\) = 6 + 18 + 54

Calculating this:

S = 6 + 18 + 54 = 78

The sum of the first three terms of the progression is \( \boxed{78} \).