<p>If 100 times the 100th term of an AP with nonzero common difference equals the 50 times its 50th term, then the 150th term of this AP is</p>
Step-by-Step Solution
Key Concept: Use the condition that 100·a₁₀₀ = 50·a₅₀ to find the relationship between first term and common difference, then determine a₁₅₀. The key is recognizing this constraint directly fixes which term equals zero.
<p><strong>Step 1:</strong> Write the general term of AP: aₙ = a + (n-1)d, where a is first term and d is common difference (d ≠ 0).</p><p><strong>Step 2:</strong> Express the given condition 100·a₁₀₀ = 50·a₅₀:<br/>100[a + 99d] = 50[a + 49d]<br/>100a + 9900d = 50a + 2450d<br/>50a + 7450d = 0<br/>a + 149d = 0</p><p><strong>Step 3:</strong> Recognize that a + 149d = 0 means:<br/>a = -149d</p><p><strong>Step 4:</strong> Find the 150th term:<br/>a₁₅₀ = a + 149d<br/>a₁₅₀ = -149d + 149d<br/>a₁₅₀ = 0</p><p><strong>Step 5:</strong> Verify: a₁₀₀ = -149d + 99d = -50d and a₅₀ = -149d + 49d = -100d<br/>100(-50d) = -5000d and 50(-100d) = -5000d ✓</p><p>∴ Answer: <strong>0</strong></p>
Correct Answer: D