<p>The sum of the series \(1 + 2 \times 3 + 3 \times 5 + 4 \times 7 + \ldots\) upto 11th term is __________.</p>
Step-by-Step Solution
Key Concept: Identify the general term as n(2n+1), then use the formula for sum of n, n² to compute Σn(2n+1) = 2Σn² + Σn efficiently rather than expanding each term.
<p><strong>Step 1:</strong> Find the general term. For position n, the term is n × (2n+1).</p><p>Check: n=1: 1×3=3 ✗ We have 1, so n=1 gives 1×1=1. Re-examine: Term 1 is 1=1×1, Term 2 is 2×3, Term 3 is 3×5, Term 4 is 4×7. Pattern: <strong>aₙ = n(2n-1)</strong></p><p><strong>Step 2:</strong> Sum up to 11 terms: S₁₁ = Σ(n=1 to 11) n(2n-1) = Σ(n=1 to 11) (2n² - n) = 2Σn² - Σn</p><p><strong>Step 3:</strong> Apply formulas: Σ(n=1 to 11) n = 11×12/2 = 66 and Σ(n=1 to 11) n² = 11×12×23/6 = 506</p><p><strong>Step 4:</strong> S₁₁ = 2(506) - 66 = 1012 - 66 = <strong>946</strong></p><p>∴ Answer: <strong>946</strong></p>
Correct Answer: 946