1²-2²+3²-4²+5^{2-..............}

^is

^1)n(n+1)/2

^2)-n(n+1)/2

^3)-n(n+2)/2

Using the identity, (a²-b²) = (a+b) * (a-b), we can write the given sequence as (assuming that the sequence has n terms) -
(1+2)(1-2) + (3+4)(3-4) + (5+6)(5-6) …....... + ((n-1)+n)((n-1)-n)
= -1(1+2) + -1(3+4) + ….... + -1(n+n-1)
= -1(1+2+3+.......+n-1+n)
And the term in the bracket is the sum of 1^stn natural numbers, and so we can write it as -
= -n(n+1)/2

Thus the answer is option 2.