Sequences & Series
Arithmetic Progression
Grade None
Question:
<p>In any A.P., if sum of first six terms is 5 times the sum of next six terms then which term is zero?</p>
<p>\(10^{\text{th}}\)</p>
<p>\(11^{\text{th}}\)</p>
<p>\(12^{\text{th}}\)</p>
<p>\(13^{\text{th}}\)</p>
Step-by-Step Solution
Key Concept: Use the formula for sum of n terms in A.P., express S₆ and S₁₂ in terms of first term and common difference, then set up the given ratio condition to solve for the term that equals zero.
<p><strong>Step 1:</strong> Let first term = a, common difference = d</p><p><strong>Step 2:</strong> Sum of first 6 terms: S₆ = 6a + 15d</p><p><strong>Step 3:</strong> Sum of first 12 terms: S₁₂ = 12a + 66d</p><p><strong>Step 4:</strong> Sum of next 6 terms (7th to 12th) = S₁₂ - S₆ = 6a + 51d</p><p><strong>Step 5:</strong> Given condition: S₆ = 5(S₁₂ - S₆)</p><p>6a + 15d = 5(6a + 51d)</p><p>6a + 15d = 30a + 255d</p><p>-24a = 240d</p><p>a = -10d</p><p><strong>Step 6:</strong> General term: aₙ = a + (n-1)d = -10d + (n-1)d = d(n - 11)</p><p><strong>Step 7:</strong> For aₙ = 0: d(n - 11) = 0, so n = 11</p><p>∴ Answer: <strong>11th term is zero</strong></p>
Correct Answer: B