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(-1)^{n}+(-1)^{2n}+(-1)^{2n+1}+(-1)^{4n+2}= -1+1-1+1=0\\ \\

reason is : 
n is a positive odd integer. So (-1)^n = -1
2n is a positive EVEN integer
.[(-1)^2]^n = 1^n = 1
2n+1 is a positive ODD integer 
(-1)^{2n+1}=[(-1)^2]^n* * (-1)^1}= 1^{n} (-1)=-1\\
4n+2 is always a positive even integer.
 (-1)^{4n+2}=[(-1)^2]^{2n+1}= 1^{2n+1}=1\\

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