Expert: Richard J. Raridon - 9/14/2010

Question
Sir, I hava a few questions in algebra imaginary numbers.

1.) 1 + i^10 + i^100 + i^1000 = 0

2.) i^104 + i^109 + i^114 + i^119 = 0

3.) 1/i - 1/i^2 + 1/i^3 - 1/i^4 = 0

4.) 2i^2 + 6i^3 + 3i^16 -6i^19 + 4i^25 = 1+4i

5.) 6i^54 + 5i^37 - 2i^11 + 6i^68 = 7i

6.) (1 + i^14 + i^18 + i^22) is a real number.

Please provide a bit explained answer and do not skip steps
as I won't be able to understand the concept and use of the
problem.

Hope you would reply soon.
                                  regards,
                                  Prakash Kumar Singh  

Answer
I assume you know that i^2 = -1, i^3 = -i, i^4 = 1, and i^5 = i
so it repeats in a cycle of 4.  
1) 1 +i^10 +i^100 +i^1000 = 1 -1 +1 +1 = 2
2) i^104 +i^109 +i^114 +i^119 = 1 +i -1 -i = 0
3) 1/i -1/i^2 +1/i^3 -1/i^4 = -i +1 +i -1 = 0
4) -2 -6i +3 +6i +4i = 1+4i
5) -6  +5i +2i +6 = 7i
6) 1 -1 -1 -1 = -2