<p>312. Let \(S_n\) denote the sum of the first \(n\) terms of an AP. If \(S_{2n} = 3S_n\), then the ratio \(S_{3n} : S_n\) is equal to</p>
Step-by-Step Solution
Key Concept: Use the formula S_n = n/2(2a + (n-1)d) to express S_{2n} and S_n in terms of a and d, then apply the condition S_{2n} = 3S_n to find the relationship between a and d. Finally, compute S_{3n}/S_n using this relationship.
<p><strong>Step 1:</strong> Apply the AP sum formula: S_n = n/2[2a + (n-1)d]</p><p>S_{2n} = 2n/2[2a + (2n-1)d] = n[2a + (2n-1)d]</p><p>S_n = n/2[2a + (n-1)d]</p><p><strong>Step 2:</strong> Use the condition S_{2n} = 3S_n:</p><p>n[2a + (2n-1)d] = 3 · n/2[2a + (n-1)d]</p><p>2[2a + (2n-1)d] = 3[2a + (n-1)d]</p><p>4a + (4n-2)d = 6a + (3n-3)d</p><p>-2a + (n+1)d = 0</p><p><strong>Step 3:</strong> Solve for the relationship: 2a = (n+1)d, so a = (n+1)d/2</p><p><strong>Step 4:</strong> Calculate S_{3n}:</p><p>S_{3n} = 3n/2[2a + (3n-1)d]</p><p>Substitute a = (n+1)d/2:</p><p>S_{3n} = 3n/2[(n+1)d + (3n-1)d] = 3n/2[4nd] = 6n²d</p><p><strong>Step 5:</strong> Calculate S_n:</p><p>S_n = n/2[2a + (n-1)d] = n/2[(n+1)d + (n-1)d] = n/2[2nd] = n²d</p><p><strong>Step 6:</strong> Find the ratio:</p><p>S_{3n} : S_n = 6n²d : n²d = 6 : 1</p><p>∴ Answer: B (6:1)</p>
Correct Answer: B