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?

Re: How to add consecutive imaginary numbers

Quote:

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?

$\displaystyle {\sum\limits_{k = 1}^N {{i^k}} = }\left\{ {\begin{array}{rl} {i,}&{N \equiv 1\;\bmod 4} \\ { - 1 + i,}&{N \equiv 2\;\bmod 4} \\ { - 1,}&{N \equiv 3\;\bmod 4} \\ {0,}&{N \equiv 0\;\bmod 4} \end{array}} \right.$

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?

Re: How to add consecutive imaginary numbers

Quote:

Originally Posted by

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

$\displaystyle \Sigma$ stands for summation.

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

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

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

Re: How to add consecutive imaginary numbers

Quote:

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!

Re: How to add consecutive imaginary numbers

Quote:

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?

$\displaystyle \sum_{k=1}^Ni^{2k-1} = \left\{ \begin{array}{rl} i & N\equiv1\mod2 \\ 0 & N\equiv0\mod2 \end{array} \right.$