To solve the problem, we need to find the sum of the first three terms of a geometric progression (G.P.) given that the first term is 6 and the fourth term is 162. Let's break this down step-by-step. ### Step 1: Understanding the G.P. In a geometric progression, each term after the first is found by multiplying the previous term by a constant called the common ratio (denoted as \( r \)). The terms of the G.P. can be expressed as: - First term (\( a_1 \)) = \( a \) - Second term (\( a_2 \)) = \( ar \) - Third term (\( a_3 \)) = \( ar^2 \) - Fourth term (\( a_4 \)) = \( ar^3 \) ### Step 2: Setting Up the Equations From the problem, we know: - \( a_1 = a = 6 \) - \( a_4 = ar^3 = 162 \) Now, substituting \( a = 6 \) into the equation for the fourth term: \[ 6r^3 = 162 \] ### Step 3: Solving for the Common Ratio \( r \) To find \( r^3 \), we can divide both sides of the equation by 6: \[ r^3 = \frac{162}{6} = 27 \] Next, we take the cube root of both sides to find \( r \): \[ r = \sqrt[3]{27} = 3 \] ### Step 4: Finding the First Three Terms Now that we have \( a = 6 \) and \( r = 3 \), we can find the first three terms of the G.P.: - First term: \( a_1 = 6 \) - Second term: \( a_2 = ar = 6 \times 3 = 18 \) - Third term: \( a_3 = ar^2 = 6 \times 3^2 = 6 \times 9 = 54 \) ### Step 5: Calculating the Sum of the First Three Terms Now, we can find the sum of the first three terms: \[ \text{Sum} = a_1 + a_2 + a_3 = 6 + 18 + 54 \] Calculating this gives: \[ \text{Sum} = 6 + 18 = 24 \] \[ \text{Sum} = 24 + 54 = 78 \] ### Conclusion: Final Answer Thus, the sum of the first three terms of the G.P. is **78**. Therefore, the correct option is **C**. ### Explanation of Other Options - **Option A (27)**: This value does not represent the sum of the first three terms. It may be a miscalculation or misunderstanding of the terms involved. - **Option B (8)**: This is too low to be the sum of the first three terms, given the values we calculated. - **Option D (48)**: This is also incorrect as it does not match the calculated sum of 78. ### Revision Summary - A geometric progression is defined by a first term and a common ratio. - The fourth term can be expressed in terms of the first term and the common ratio. - Solving for the common ratio involves manipulating the equation derived from the terms. - The sum of the first three terms can be calculated directly once the terms are known. This thorough breakdown should help you understand how to approach similar problems involving geometric progressions in the future!