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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]


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Solution

Proof of the theorem [x]+[x+1/n]+[x+2/n]+.+[x+(n-2)/n]+[x+(n-2)/n][x+(n-1)/n]=[nx]

Since, the greatest integer is defined as follow:

[m+(p/q)]=m

Where mis an integer value.

(p/q)is the fractional value less than 1

Now,

[x]+[x+1/n]+[x+2/n]+.+[x+(n-2)/n]+[x+(n-2)/n][x+(n-1)/n]=x+x+x....ntimes[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] is proved

Hence, The theorem is proved.


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