{an}: -1, 3, -6, 10, -15, 21 ....?

Double them and get

{2an}: -2, 6, -12, 20, -30, 42 ....

Write them as the product of two integers:

{2an}: -1×2, 2×3, -3×4, 4×5, -5×6, 6×7 ....

To get the alternating signs we use the factor (-1)n×

{2an}: (-1)1×1×2, (-1)2×2×3, (-1)3×3×4, (-1)4×4×5, (-1)5×5×6, (-1)6×6×7 ....

So the general term for 2an is  

2an (-1)n×n×(n+1)

So the general term for an is half that, so
we divide both sides by 2:

     (-1)n×n×(n+1)
an = —————————————
           2

or eliminating the multiplication symbols:

     (-1)nn(n+1)
an = ———————————
          2

Edwin