Question 1046622
gauss's trick is described here.


<a href = "https://nzmaths.co.nz/gauss-trick-staff-seminar" target = "_blank">https://nzmaths.co.nz/gauss-trick-staff-seminar</a>


i used his method to determine the value of the sum of the odd number from 1 to 999.


the sum of the number from 1 to 1000 can be found as follows:


in normal order, the sequence is 1,2,3,4,5.
in reverse order, the sequence is 1000,999,998,997,994
pair the normal order terms with the reverse order terms and you get:
1 + 1000 = 1001
2 + 999 = 1001
3 + 998 = 1001
etc.
there are 1000 of such pairings.
1000 * 1001 = 1001000.
that's double the sum of 1 to 1000.
divide that by 2 and you get the sum of the numbers from 1 to 1000 = 500500.


there are 500 pairs of numbers from 1 to 1000.
the pairs are:
1,2
3,4
5,6
etc.
in each pair, the even number is 1 more than the odd number.


since there are 500 pairs, then the sum of the even numbers will be 500 more than the sum of the odd numbers.


let x = the sum of the odd number.
x + 500 is the sum of the even numbers.


you get x + x + 500 = 500500
simplify to get 2x + 500 = 500500
subtract 500 from 500500 to get:
2x = 500000
divide both sides by 2 to get:
x = 250000
that's the sum of the odd numbers.
the sum of the even numbers is 250500
add them up and you get the sum of the number is 500500.


i check with excel and got those tallies, so the formula works.


the sum of the odd number is 250,000.
the sum of the even numbers is 250,500.


here's a picture of my excel analysis.


<img src = "http://theo.x10hosting.com/2016/090603.jpg" alt="$$$" </>