what is the proof of the theorem:

[x]+[x+1/n]+[x+2/n]+…….+[x+(n-2)/n]+[x+(n-2)/n][x+(n-1)/n] =[nx]

We know that gratest integer possese property as mentioned below

[m + (p/q)] m

where

m = integer

(p/q) = fractional value <1

So we can write

[x]+[x+1/n]+[x+2/n]+…….+[x+(n-2)/n]+[x+(n-2)/n][x+(n-1)/n] as

x + x +x……..till n times

[x]+[x+1/n]+[x+2/n]+…….+[x+(n-2)/n]+[x+(n-2)/n][x+(n-1)/n] = n*x

∴ [x]+[x+1/n]+[x+2/n]+…….+[x+(n-2)/n]+[x+(n-2)/n][x+(n-1)/n] = [nx]

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