Difference of positive and negative integers

[westworld]

Find the difference between the sum of the first 500 even positive integers and the first 500 odd positive integers.

 

[pka]

Find the difference between the sum of the first 500 even positive integers and the first 500 odd positive integers.

You need to show some of your own effort.
Hint: \(\displaystyle \sum\limits_{k = 1}^{500} {2k} - \sum\limits_{k = 1}^{500} {\left( {2k - 1} \right)}=~? \)

 

[soroban]

Hello, westworld!

Find the difference between the sum of the first 500 even positive integers
and the first 500 odd positive integers.

Here is a truly childish way to solve it . . .


\(\displaystyle X \;=\;(2 + 4 + 6 + 8 + \hdots + 1000) - (1 + 3 + 5 + 7 + \hdots + 999)\)

. . .\(\displaystyle =\;(2-1) + (4-3) + (6-5) + (8-7) + \hdots + (1000-999)\)

. . .\(\displaystyle =\;\underbrace{1 + 1 + 1 + 1 + \hdots + 1}_{\text{How many 1's are there?}}\)

 

[lookagain]

Here is a truly \(\displaystyle > > \)childish \(\displaystyle < < \)way to solve it . . .


\(\displaystyle X \;=\;(2 + 4 + 6 + 8 + \hdots + 1000) - (1 + 3 + 5 + 7 + \hdots + 999)\)

. . .\(\displaystyle =\;(2-1) + (4-3) + (6-5) + (8-7) + \hdots + (1000-999)\)

. . .\(\displaystyle =\;\underbrace{1 + 1 + 1 + 1 + \hdots + 1}_{\text{How many 1's are there?}}\)

I disagree.

I would call it "childlike" instead of "childish."

And I like it!

 

[westworld]

dIFFERENCE

i GET THE DIFFERENCE TO BE 500.

 

[srmichael]

Find the difference between the sum of the first 500 even positive integers and the first 500 odd positive integers.

This was an actual problem on a past SAT exam, except 500 was 100 in the problem. Not a big fan of the Math portion of the SAT because of problems like these. The math portion of the ACT is much more direct, true math kind of problems, without quirky problems like this one.