Question:
If \([x]\) be the greatest integer less than or equal to \(x\), then \(\sum_{n=8}^{100}\left[\frac{(-1)^n n}{2}\right]\) is equal to :
If \([x]\) be the greatest integer less than or equal to \(x\), then \(\sum_{n=8}^{100}\left[\frac{(-1)^n n}{2}\right]\) is equal to :
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Separate even and odd \(n\): for odd \(n\) the floor of a negative half-integer rounds down.
Updated On: Nov 4, 2026
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The Correct Option is B
Solution and Explanation
Even \(n=2m\) (\(m=4,\dots,50\)): \(\left[\frac{2m}{2}\right]=m\).
Odd \(n=2m-1\) (\(m=5,\dots,50\)): \(\left[-\frac{2m-1}{2}\right]=[-m+\tfrac12]=-m\).
Sum \(=\sum_{m=4}^{50}m-\sum_{m=5}^{50}m=4\).
Answer: (2) 4
Odd \(n=2m-1\) (\(m=5,\dots,50\)): \(\left[-\frac{2m-1}{2}\right]=[-m+\tfrac12]=-m\).
Sum \(=\sum_{m=4}^{50}m-\sum_{m=5}^{50}m=4\).
Answer: (2) 4
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