If [x] denotes the greatest integer less than | vedclass

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(B) Let $S = \sum_{n=8}^{100} \left[ \frac{(-1)^{n} n}{2} \right]$.Expanding the sum,we get:$S = \left[ \frac{8}{2} \right] + \left[ \frac{-9}{2} \rig

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$4$

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(B) Let $S = \sum_{n=8}^{100} \left[ \frac{(-1)^{n} n}{2} \right]$.Expanding the sum,we get:$S = \left[ \frac{8}{2} \right] + \left[ \frac{-9}{2} \right] + \left[ \frac{10}{2} \right] + \left[ \frac{-11}{2} \right] + \dots + \left[ \frac{-99}{2} \right] + \left[ \frac{100}{2} \right]$.Using the property $[x]$,we have:$S = 4 + [-4.5] + 5 + [-5.5] + 6 + [-6.5] + \dots + [-49.5] + 50$.$S = 4 + (-5) + 5 + (-6) + 6 + (-7) + \dots + (-50) + 50$.Notice that the terms cancel out in pairs: $(-5+5) + (-6+6) + \dots + (-50+50) = 0$.Therefore,$S = 4 + 0 = 4$.

If $[x]$ denotes 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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