Expert: Sherman D. - 5/12/2005

Question
1) show that (-1)^1 + (-1)^2 + (-1)^3 + ... + (-1)^n= ((-1)^n-1)/2

Answer
if "n" is even both sides will equal 0.

for ex: n = 12
(-1)^1 + (-1)^2 + (-1)^3 + (-1)^4 + (-1)^5 + (-1)^6 +
(-1)^7 + (-1)^ + (-1)^9 + (-1)^10 + (-1)^11 + (-1)^12 = ((-1)^(12) - 1)/2

-1 + 1 - 1 + 1 - 1 + 1 - 1 + 1 - 1 + 1 - 1 + 1 = (1 - 1)/2
0 + 0 + 0 + 0 + 0 + 0 = 0/2 = 0

if "n" is an odd number, then both sides will equal -1, or think of it as even--which equals 0-- + (-1).

for ex:

n = 11

(-1)^1 + (-1)^2 + (-1)^3 + (-1)^4 + (-1)^5 + (-1)^6 +
(-1)^7 + (-1)^ + (-1)^9 + (-1)^10 + (-1)^11 = ((-1)^(11) - 1)/2

-1 + 1 - 1 + 1 - 1 + 1 - 1 + 1 - 1 + 1 - 1 =
0 + 0 + 0 + 0 + 0 - 1 = (-1 - 1)/2 = (-2)/2 = -1

so even = 0, odd = -1

Sorry i couldn't give you a better explaination.