What is the sum of all proper fractions with denominator less than 2016?

The answer is 1014552.5.

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Let's start at this one,

d < 2016In this case, we have 2015, 2014, 2013, ... , 4, 3, 2 as denominator (1 is not included since we are talking about proper fraction.

Let's find first the sum of all proper fraction with denominator 2015, 2014, 2013 respectively, then

$\frac{1}{2015}$ + $\frac{2}{2015}$ + $\frac{3}{2015}$ + ... + $\frac{2013}{2015}$ + $\frac{2014}{2015}$ = $\frac{\frac{(2014)(2015)}{2}}{2015}$ =

1007$\frac{1}{2014}$ + $\frac{2}{2014}$ + $\frac{3}{2014}$ + ... + $\frac{2012}{2014}$ + $\frac{2013}{2014}$ = $\frac{\frac{(2013)(2014)}{2}}{2014}$ =

1006.5$\frac{1}{2013}$ + $\frac{2}{2013}$ + $\frac{3}{2013}$ + ... + $\frac{2011}{2013}$ + $\frac{2012}{2013}$ = $\frac{\frac{(2012)(2013)}{2}}{2013}$ =

1006Notice that the sum obtained of each common difference of -

0.5. UsingArithmetic Sumof $S_{n}$ = $\frac{n}{2}$ ( $a_{1}$ + $a_{n}$ ) with last term0.5( sum of proper fraction with denominator 2) andn = 2014(number of terms)Then we have $S_{2014}$ = $\frac{2014}{2}$ ( 0.5 + 1007 ) = (1007)( $\frac{2015}{2}$ ) =

1014552.5Hence the sum of all proper fraction less than 2016 is

$\boxed{1014552.5}$