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prove:- [x/n]+[x+1/n]+.................+[x+n-1/n]=[x] where n belongs to N

prove:-


[x/n]+[x+1/n]+.................+[x+n-1/n]=[x]


where n belongs to N

Grade:11

1 Answers

Jitender Singh IIT Delhi
askIITians Faculty 158 Points
9 years ago
Ans:
[\frac{x}{n}]+[\frac{x+1}{n}]+[\frac{x+2}{n}] + .......... + [\frac{x+n-1}{n}] = [x]
Let
[x] = nN
nN<x<nN+1
N<\frac{x}{n}<N+\frac{1}{n}
N+\frac{1}{n}<\frac{x+1}{n}<N+\frac{2}{n}
N+\frac{2}{n}<\frac{x+2}{n}<N+\frac{3}{n}
Similarly upto nth term,
N+\frac{n-1}{n}<\frac{x+n-1}{n}<N+1
[\frac{x}{n}]+[\frac{x+1}{n}]+[\frac{x+2}{n}] + .......... + [\frac{x+n-1}{n}] = N + N+........N = nN = [x]
Thanks & Regards
Jitender Singh
IIT Delhi
askIITians Faculty

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