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Bipin Dubey (13679)

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Sum is [a/b] + [2a/b] +........+[(b-2)a/b] + [(b-1)a/b]

Let a/b = x.y where x is its integral part and y be its fraction
Take 1st and last term
1st term = [a/b] = x
last term = [(b-1)a/b] = [a - (a/b)]
                               = [a - x.y]
                               = [a - x - 0.y]
                               = a - x - 1        (since a and x are integers and y is fraction)

hence 1st term + last term = x + a - x - 1
                                        = a - 1
similarly 2nd term + second last term = a - 1
similarly for the other terms

Since there are b-1 numbers hence they would form (b-1)/2 pairs of a - 1
hence when (a-1) is added (b-1)/2 times it gives

(a-1)(b-1)/2

I think its clear now
Best Wishes

Bipin Kumar Dubey Chemical Dept. IIT Kharagpur

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