<p>If \(t_n\) denotes the \(n\)th term of the series \(2 + 3 + 6 + 11 + 18 + \cdots\) then \(t_{50}\) is</p>
Step-by-Step Solution
Key Concept: Recognize this as a second-order difference sequence by computing successive differences: first differences are 1, 3, 5, 7,... (arithmetic), indicating the nth term follows a quadratic formula.
<p><strong>Step 1:</strong> Find the pattern by computing differences.</p><p>Sequence: 2, 3, 6, 11, 18, ...</p><p>First differences: 1, 3, 5, 7, ... (arithmetic with common difference 2)</p><p>Second differences: 2, 2, 2, ... (constant)</p><p><strong>Step 2:</strong> Since second differences are constant, tn is quadratic. Let tn = an² + bn + c.</p><p>From t₁ = 2: a + b + c = 2</p><p>From t₂ = 3: 4a + 2b + c = 3</p><p>From t₃ = 6: 9a + 3b + c = 6</p><p><strong>Step 3:</strong> Solve the system.</p><p>Subtracting equation 1 from 2: 3a + b = 1</p><p>Subtracting equation 2 from 3: 5a + b = 3</p><p>Subtracting these: 2a = 2 → a = 1</p><p>Therefore: b = 1 - 3(1) = -2, and c = 2 - 1 + 2 = 3</p><p>So tn = n² - 2n + 3</p><p><strong>Step 4:</strong> Calculate t₅₀.</p><p>t₅₀ = (50)² - 2(50) + 3 = 2500 - 100 + 3 = 2403</p><p><strong>Verification:</strong> t₁ = 1 - 2 + 3 = 2 ✓, t₂ = 4 - 4 + 3 = 3 ✓</p><p>∴ Answer: 2403</p>
Correct Answer: D