Question 744746
The sum of -39 ...+39 = 0.  -39 + 39 - 0.  -37 + 37 = 0 etc.<P>
This is the sum of odd integers from 41 to 91.<P>
The sum of n odd integers is {{{n^2}}}<P>
The number, n, of odd positive integers from 0 up to a certain odd integer is ((that integer - 1)/2) + 1.  For example, the number (n) of odd integers from 0 to 9 is ((9-1)/2) + 1 = 5.  The odd integers from 0 to 9 are 1, 3, 5, 7, 9 - there are five.<P>
The number of odd integers from 0 to 91 is 91-1 = 90/2=45 + 1 = 46.  {{{46^2 = 2116}}}.<P>
Subtract the sum of odd integers from 0 to 39 (because those summed to 0 when added to -1, -3...-39).  There are (39-1)/2 = 19 + 1 = 20 odd integers from 0 to 37.  {{{20^2 = 400}}}<P>
The sum of all odd numbers from 0 to 91 - the sum of all odd numbers from 0 to 39 is the sum of odd numbers from 41 to 91.  It is 2116-400 = 1716.