Question

show whether the statement is true:
for all whole numbers n, (-1)^2n =1 and (-1)^2n+1 = -1

Answer 1

for the first one:
(-1)^(2n) = +1
2n is an even number, so when you multiply -1 an even number of times you get a positive
e.g.
(-1)^12= ( (-1)^2 )^6
= 1^6 = 1

but (-1)^13
= (-1)^12 (-1)
= (+1)(-1) = -1

so if you raise (-1)^odd, you get -1

which answers the second part since
2n+1 is always an odd number

be aware of the difference between something like
(-1)^12 and -1^12 ,
(-1)^even = +1

but
-1^even = -1
-1^odd = +1