Find the sum of 12 - 22 + 32 - 42 + 52 - 62........?

The nth term is =(−1)n+1n2

The sum is S=12−22+32−42+52−62+……..+(n−1)2−n2, ∀n ∈ N

If n is even

S=(12−22)+(32−42)+(52−62)+……+((n−1)2−n2)

S=(1−2)(1+2)+(3−4)(3+4)+(5−6)(5+6)+…+((n−1−n)(n−1+n)

S=(−1)(1+2)+(−1)(3+4)+(−1)(5+6)+….+((−1)(n−1+n)

S=(−1)((1+2)+(3+4)+(5+6)+….+(n−1)+n)

=(−1)⋅n(1+n)/2

=(−1)n(n+1)/2

If n is odd

S=(−1)n+1n(n+1)/2

The sum of the series is =(−1)n+1n(n+1)/2 = (−1)20+120(20+1)/2 = -210

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