1^2 - 2^2 + 3^2 - 4^2 + ..... - 2012^2 + 2013^2
Since the sequence stops at 2013 and doesn't go on to 2014, I'll assume it was
last year's problem. However, the answer would have been negative if they
had made it go to -2014. LOL
(You can make 2 exponents ² by typing 253 on the keypad at the far right side of
your keyboard while holding down the ALT key. (It doesn't work if you type 253
on the number keys above the letter keys. You must use the numbers on the
keypad on the far right). You can make cubes ³ if you use 1079, but
unfortunately no higher exponents)
I'll put in a few more terms:
1²-2²+3²-4²+5²-6²+7²+··· -2010²+2011²-2012²+2013²
After the first term, swap 2nd and 3rd terms, 5th and 4th terms, etc., ...,
2012th and 2013th.
1²+3²-2²+5²-4²+7²-6²+···+2011²-2010²+2013²-2012²
I did that to make differences of successive terms positive, so I could group
them into differences of successive squares like this:
1²+(3²-2²)+(5²-4²)+(7²-6²)+···+(2011²-2010²)+(2013²-2012²)
Now we can factor each of those terms in parentheses as the difference of two
squares:
1²+(3-2)(3+2)+(5-4)(5+4)+(7-6)(7+6)+···+(2011-2010)(2011+2010)+(2013-2012)(2013+2012)
which becomes
1+(1)(5)+(1)(9)+(1)(13)+···+(1)(4021)+(1)(4025)
or
1+5+9+13+···+4023+4025
That is an arithmetic sequence with first term 
and common difference d=4,
and 
.
Edwin
