1. ## How to add consecutive imaginary numbers

What is i + i^2 + i^3 + ... + i^49?

I know for each one there is only 4 possibilities of solutions, i, -i, 1, and -1. So I divided 49 by 4 and got 12.25 so there's 12 of each solution plus one extra one. So I did: 12i + -12i + 12 + -12 and then the 49th term I think is i but I'm not sure because I forgot what the value is (calculator is not allowed). So everything cancels out and I'm just left with i as the last term...But I don't think the solution is i because I don't think I did it correctly.

What if I were told to add up every other consecutive one starting from i? Like i + i^3 + i^5 + ... i^49?

2. ## Re: How to add consecutive imaginary numbers

Originally Posted by daigo
What is i + i^2 + i^3 + ... + i^49?
I know for each one there is only 4 possibilities of solutions, i, -i, 1, and -1. So I divided 49 by 4 and got 12.25 so there's 12 of each solution plus one extra one. So I did: 12i + -12i + 12 + -12 and then the 49th term I think is i but I'm not sure because I forgot what the value is (calculator is not allowed). So everything cancels out and I'm just left with i as the last term...But I don't think the solution is i because I don't think I did it correctly.
What if I were told to add up every other consecutive one starting from i? Like i + i^3 + i^5 + ... i^49?
${\sum\limits_{k = 1}^N {{i^k}} = }\left\{ {\begin{array}{rl} {i,}&{N \equiv 1\;\bmod 4} \\ { - 1 + i,}&{N \equiv 2\;\bmod 4} \\ { - 1,}&{N \equiv 3\;\bmod 4} \\ {0,}&{N \equiv 0\;\bmod 4} \end{array}} \right.$

3. ## Re: How to add consecutive imaginary numbers

Sorry, what does that 'E' symbol mean and what are the variables equal to? And the 'mod 4' part?

4. ## Re: How to add consecutive imaginary numbers

Originally Posted by daigo
Sorry, what does that 'E' symbol mean and what are the variables equal to? And the 'mod 4' part?
$\Sigma$ stands for summation.

${\sum\limits_{k = 1}^6 {{i^k}} = }i+i^2+i^3+i^4+i^5+i^6=-1+i$

${N \equiv j\;\bmod 4}$, $~j$ is the reminder when $N$ is divided by $4$.

So, ${49 \equiv 1\;\bmod 4}$ thus ${\sum\limits_{k = 1}^{49} {{i^k}} = }i$

5. ## Re: How to add consecutive imaginary numbers

Originally Posted by daigo
Sorry, what does that 'E' symbol mean and what are the variables equal to? And the 'mod 4' part?
If it helps:

you can think of clocks being mod 12. We know that 13:00 is 1pm, 16:00 is 4pm etc., We say that the numbers are equal if they differ by 12. So 13 = 1 mod 12, and 16=4 mod 12.

Similarly, when working with degrees often we operate mod 360. So you would say that 380 degrees = 20 degrees. This is because 380=20 mod 360!

6. ## Re: How to add consecutive imaginary numbers

Originally Posted by daigo
What if I were told to add up every other consecutive one starting from i? Like i + i^3 + i^5 + ... i^49?
$\sum_{k=1}^Ni^{2k-1} = \left\{ \begin{array}{rl} i & N\equiv1\mod2 \\ 0 & N\equiv0\mod2 \end{array} \right.$