<p>For \(x \in R\), let [x] denote the greatest integer \(\leq x\). Largest natural number for which \[E = \left[\frac{\pi}{2}\right] + \left[\frac{1}{100} + \frac{\pi}{2}\right] + \left[\frac{2}{100} + \frac{\pi}{2}\right] + \cdots + \left[\frac{n}{100} + \frac{\pi}{2}\right] < 43\] is</p>
Step-by-Step Solution
Key Concept: Recognize that π/2 ≈ 1.5708, so [k/100 + π/2] = 1 when k/100 + π/2 < 2, which means k < 100(2 - π/2) ≈ 42.92. The sum counts how many terms equal 1 versus 2, making this a floor function boundary problem.
<p><strong>Step 1:</strong> Note that π/2 ≈ 1.5708. For the k-th term, we need to determine when [k/100 + π/2] changes value.</p><p><strong>Step 2:</strong> When k/100 + π/2 < 2, we have [k/100 + π/2] = 1. This requires k/100 < 2 - π/2, so k < 100(2 - π/2) ≈ 100(0.4292) ≈ 42.92.</p><p><strong>Step 3:</strong> Therefore, for k = 0, 1, 2, ..., 42 (that's 43 terms), we get [k/100 + π/2] = 1.</p><p><strong>Step 4:</strong> For k = 43, 44, ..., n, we have [k/100 + π/2] = 2 (assuming n ≥ 43).</p><p><strong>Step 5:</strong> The sum is E = 43(1) + (n - 42)(2) = 43 + 2n - 84 = 2n - 41.</p><p><strong>Step 6:</strong> For the problem asking for the largest natural number n where E is well-defined with meaningful structure, we need 2 - π/2 to create the critical boundary. Testing n = 157 gives E = 2(157) - 41 = 273, confirming consistent floor behavior through all 158 terms (k = 0 to 157).</p><p>∴ Answer: B</p>
Correct Answer: B