To find the common ratio of the geometric progression (GP) given the three consecutive terms \( n-2 \), \( n \), and \( n+3 \), we need to understand the properties of a GP. In a GP, the ratio of any two consecutive terms is constant, which means: \[ \text{Common Ratio} = \frac{\text{Second Term}}{\text{First Term}} = \frac{\text{Third Term}}{\text{Second Term}} \] ### Step-by-Step Explanation 1. **Set Up the Ratios**: We can express the common ratio \( r \) as follows: \[ r = \frac{n}{n-2} = \frac{n+3}{n} \] 2. **Cross-Multiply to Eliminate the Fractions**: From the first ratio: \[ r(n-2) = n \quad \Rightarrow \quad rn - 2r = n \quad \Rightarrow \quad rn - n = 2r \quad \Rightarrow \quad n(r-1) = 2r \] From the second ratio: \[ r(n) = n + 3 \quad \Rightarrow \quad rn = n + 3 \quad \Rightarrow \quad rn - n = 3 \quad \Rightarrow \quad n(r-1) = 3 \] 3. **Equate the Two Expressions for \( n(r-1) \)**: Now we have two equations: \[ n(r-1) = 2r \quad \text{(1)} \] \[ n(r-1) = 3 \quad \text{(2)} \] Setting these equal gives: \[ 2r = 3 \quad \Rightarrow \quad r = \frac{3}{2} \] 4. **Conclusion**: The common ratio \( r \) is \( \frac{3}{2} \). ### Why the Other Options Are Incorrect - **Option A: \( \frac{1}{4} \)**: If \( r = \frac{1}{4} \), substituting back into the equations would not satisfy the equality \( n(r-1) = 2r \) or \( n(r-1) = 3 \). This would lead to contradictions. - **Option B: \( \frac{1}{2} \)**: Similar to option A, substituting \( r = \frac{1}{2} \) into the equations would not yield valid values for \( n \) that satisfy both equations. - **Option C: \( \frac{2}{3} \)**: Again, substituting \( r = \frac{2}{3} \) would not satisfy the derived equations, leading to inconsistencies. ### Common Pitfalls - **Misunderstanding the GP Property**: Remember that in a GP, the ratio between consecutive terms must be equal. This is crucial for setting up the equations correctly. - **Algebraic Manipulation Errors**: Be careful when cross-multiplying and rearranging equations. Small mistakes can lead to incorrect conclusions. ### Revision Summary - The common ratio in a geometric progression is the ratio of any two consecutive terms. - Set up equations based on the property of the GP and solve for the common ratio. - Ensure to check the derived common ratio against the original terms to confirm validity. - Be cautious with algebraic manipulations to avoid errors. The correct answer is **D. \( \frac{3}{2} \)**.