<p>Let \(f(n) = \left[\dfrac{1}{3} + \dfrac{3n}{100}\right]n\), where \([n]\) denotes the greatest integer less than or equal to \(n\). Then \(\displaystyle\sum_{n=1}^{56} f(n)\) is equal to</p>
Step-by-Step Solution
Key Concept: Split the sum based on when the floor function value changes by analyzing ⌊1/3 + 3n/100⌋ for different ranges of n. The floor value increases at specific thresholds determined by 1/3 + 3n/100 crossing integers.
<p><strong>Step 1:</strong> Determine when ⌊1/3 + 3n/100⌋ changes value.</p><p>For the floor value to equal k: k ≤ 1/3 + 3n/100 < k+1</p><p>This gives: (100k - 100/3)/3 ≤ n < (100k + 200/3)/3</p><p><strong>Step 2:</strong> Find the ranges:</p><p>• When k=0: n < 33.33..., so n ∈ {1,2,...,33} but checking: 1/3 + 3(33)/100 = 1/3 + 0.99 ≈ 1.32, so k=1 for n≥23</p><p>• When k=0: 1/3 + 3n/100 < 1 ⟹ n < 100/3(1 - 1/3) = 66.67, and 1/3 + 3n/100 ≥ 0 ⟹ n ≥ -11.11</p><p>• ⌊1/3 + 3(22)/100⌋ = ⌊0.993...⌋ = 0</p><p>• ⌊1/3 + 3(23)/100⌋ = ⌊1.023...⌋ = 1</p><p>• ⌊1/3 + 3(56)/100⌋ = ⌊1.68...⌋ = 1</p><p><strong>Step 3:</strong> Partition the sum:</p><p>∑(n=1 to 56) f(n) = ∑(n=1 to 22) 0·n + ∑(n=23 to 56) 1·n</p><p>= 0 + ∑(n=23 to 56) n</p><p>= ∑(n=1 to 56) n - ∑(n=1 to 22) n</p><p>= [56(57)/2] - [22(23)/2]</p><p>= 1596 - 253 = 1343</p><p>∴ Answer: <strong>1343</strong></p>
Correct Answer: D