Sequences & Series
Arithmetic Progression
Grade 11
Question:
<p>If the arithmetic progression whose common difference is non-zero, the sum of first \(3n\) terms is equal to the sum of the next \(n\) terms. The ratio of the sum of the first \(2n\) terms to the next \(2n\) terms is</p>
<p>(a) \(\frac{1}{5}\)</p>
<p>(b) \(\frac{2}{3}\)</p>
<p>(c) \(\frac{3}{4}\)</p>
<p>(d) None of these</p>
Step-by-Step Solution
Key Concept: Use the sum formula for AP to express the given condition (sum of first 3n terms equals sum of next n terms), then solve for the relationship between first term and common difference to find the required ratio.
<p><strong>Step 1: Set up the given condition.</strong></p><p>Let the AP have first term <em>a</em> and common difference <em>d</em> (where d ≠ 0).</p><p>Sum of first 3n terms: <em>S</em><sub>3n</sub> = (3n/2)[2a + (3n-1)d]</p><p>Sum of next n terms (terms from 3n+1 to 4n): = <em>S</em><sub>4n</sub> - <em>S</em><sub>3n</sub></p><p>where <em>S</em><sub>4n</sub> = (4n/2)[2a + (4n-1)d] = 2n[2a + (4n-1)d]</p><p><strong>Step 2: Express the given condition.</strong></p><p>S<sub>3n</sub> = S<sub>4n</sub> - S<sub>3n</sub></p><p>2S<sub>3n</sub> = S<sub>4n</sub></p><p>2 · (3n/2)[2a + (3n-1)d] = 2n[2a + (4n-1)d]</p><p>3n[2a + (3n-1)d] = 2n[2a + (4n-1)d]</p><p><strong>Step 3: Simplify.</strong></p><p>3[2a + (3n-1)d] = 2[2a + (4n-1)d]</p><p>6a + 3(3n-1)d = 4a + 2(4n-1)d</p><p>6a + (9n-3)d = 4a + (8n-2)d</p><p>2a + (n-1)d = 0</p><p>2a = -(n-1)d</p><p><strong>Step 4: Find the required ratio.</strong></p><p>Sum of first 2n terms: S<sub>2n</sub> = (2n/2)[2a + (2n-1)d] = n[2a + (2n-1)d]</p><p>Sum of next 2n terms (terms from 2n+1 to 4n): = S<sub>4n</sub> - S<sub>2n</sub></p><p>S<sub>4n</sub> = 2n[2a + (4n-1)d]</p><p>Next 2n terms = 2n[2a + (4n-1)d] - n[2a + (2n-1)d]</p><p>= n{2[2a + (4n-1)d] - [2a + (2n-1)d]}</p><p>= n{4a + (8n-2)d - 2a - (2n-1)d}</p><p>= n{2a + (6n-1)d}</p><p><strong>Step 5: Substitute the relation 2a = -(n-1)d.</strong></p><p>S<sub>2n</sub> = n[2a + (2n-1)d] = n{-(n-1)d + (2n-1)d} = n{-(n-1)d + (2n-1)d} = n{d(n)d} = n²d</p><p>Next 2n terms = n{2a + (6n-1)d} = n{-(n-1)d + (6n-1)d} = n{d(5n)} = 5n²d</p><p><strong>Step 6: Calculate the ratio.</strong></p><p>Ratio = S<sub>2n</sub> / (Next 2n terms) = n²d / 5n²d = 1/5</p><p><strong>∴ Answer: A</strong></p>
Correct Answer: A