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Prove the following equation12-22+32-42+…+(-1) n-1N2 = (-1) n-1.
Test answer: Solution: (1) When n= 1, the left side = 12= 1.

Right = (- 1) 0

The original equation holds.

(2) Assuming that n=k(k∈N*, k≥ 1), the equation holds.

That is,12-22+32-42+…+(-1) k-1k2 = (-1) k-1.

Then, when n=k+ 1

12-22+32-42+…+(- 1)k- 1 k2+(- 1)k(k+ 1)2

=(- 1)k- 1

+(- 1)k (k+ 1)2

When n = k+ 1, the equation also holds.

From (1)(2), any n∈N* has.

12-22+32-42+…+(- 1)n- 1 N2 =(- 1)n- 1