Does (1+2)(1-2)+(3+4)(3-4)+...+(2007+2008)(2007-2008) equal -(1+2+3+...+2008)?

[Kartik.]

S = 1²-2²+3²-4²...+2009²

Attempt=

S = (1+2)(1-2)+(3+4)(3-4)+...+(2007-2008)(2007+2008) [can we write this as -(1+2+3+4+5...2008) if yes, then why ?) +2009²
Stuck after this.

 

[Saitama]

Do you know the summation of:
1²+2²+3²+4²+...+n²

That would be of help here.

 

[Mentallic]

S = 1²-2²+3²-4²...+2009²

Attempt=

S = (1+2)(1-2)+(3+4)(3-4)+...+(2007-2008)(2007+2008) [can we write this as -(1+2+3+4+5...2008) if yes, then why ?) +2009²
Stuck after this.

Yes you can write it as -(1+2+...+2008)+2009² because

(1+2)(1-2) = -(1+2)
(3+4)(3-4) = -(3+4)
...

And finally, we factored 2007²-2008² but left out the 2009² term so we need to add that term in at the end.

Now, do you know the formula to sum the first n natural numbers?

Hint:

1+2+3+...+(n-2)+(n-1)+n

= (1+n) + (2+(n-1)) + (3+(n-2))+...

 

[Bonaparte]

Just to give you a greater hint to support Mentallic, draw out the numbers from 1 to n
1+2+3+4...n-3+n-2+n-1+n, now notice, like Mentallic said, how 1+n = n-1+2 = n-2+3, so all those pairs have the same value, and when you have x things with value y, the result is...
Bonaparte