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 \( \boxed{78} \).