What is 1*2-2*3+3*4-4*5+5*6-6*7+...+2015*2016-2016*2017=?

Question

I haven't learned about shortes way to solve this so please help. I am serious.

Answer 1

1*2-2*3+3*4-4*5+5*6-6*7+...+2015*2016-2016*2017=?

Can we re-write this series??

Let’s extract pairs of terms with even factors.  For example:

      5*6 – 6*7

        6*(5 – 7)

           -12

   And

       6*7 – 7*8

       7*(6 – 8)

           -14

 

Aha!

Taking two terms at a time, this series reduces to:

     - 4  - 8 - 12 - … - 4032          [note:   (-2)*2016]

 

Further, let’s factor out (-4) and make the series:

    (-4)* (1 + 2 + 3 + … + 1008)       [note:  there are 1008 terms]

 

Now, remember Gauss’s formula for the sum of the numbers from 1 to n:

         n*(n+1)/2

 

The sum is:

     (-4)*( 1008*1009/2 ) = -2034144

 

p.s., Most complex problems can be solved when they are merely a combination of simple problems (it's called "divide and conquer").