For x∈R, let [x] denotes the greatest integer ≤ x, then the value of −$$13+$$−$$13$$−$$1100+$$−$$13$$−$$2100+$$,…,$$+$$−$$13$$−$$99100$$ is

Question

For x∈R, let [x] denotes the greatest integer ≤ x, then the value of −$$13+$$−$$13$$−$$1100+$$−$$13$$−$$2100+$$,…,$$+$$−$$13$$−$$99100$$ is

Infinity Learn · Accepted Answer

For $$0$$≤r≤$$66$$,$$0$$≤$$r100$$<$$23$$⇒ −$$23$$<−$$r100$$≤$$0$$⇒ −$$13$$−$$23$$<−$$13$$−$$r100$$≤−$$13$$∴ −$$13$$−$$r100=$$−$$1$$ for $$0$$≤r≤$$66Also$$, for $$67$$≤r≤$$100$$,$$67100$$≤$$r100$$≤$$1$$⇒ −$$1$$≤−$$r100$$≤−$$67100$$⇒ −$$13$$−$$1$$≤−$$13$$−$$r100$$≤−$$13$$−$$67100$$∴ −$$13$$−$$r100=$$−$$2$$ for $$67$$≤r≤$$100Hence$$, ∑$$r=0100$$ −$$13$$−$$r100=67($$−$$1)+2($$−$$34)=$$−$$135$$

For $x \in R$ , let [x] denotes the greatest integer ≤ x, then the value of $(- \frac{1}{3}) + (- \frac{1}{3} - \frac{1}{100}) + (- \frac{1}{3} - \frac{2}{100}) +, \dots, + (- \frac{1}{3} - \frac{99}{100})$ is

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For x∈R, let [x] denotes the greatest integer ≤ x, then the value of −13+−13−1100+−13−2100+,…,+−13−99100 is

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For $x \in R$ , let [x] denotes the greatest integer ≤ x, then the value of $(- \frac{1}{3}) + (- \frac{1}{3} - \frac{1}{100}) + (- \frac{1}{3} - \frac{2}{100}) +, \dots, + (- \frac{1}{3} - \frac{99}{100})$ is