I'd like to form the magnitude and phase of the following complex number:
$$[1 + exp(i\theta) + exp(i2\theta) + exp(i3\theta) + exp(i4\theta)]$$
I was told to take the $exp(i2\theta)$ outside and simplify using Euler's formula, but I've reached a block.
$$\Rightarrow exp(i2\theta)[exp(-i2\theta) + exp(-i\theta) + 1 + exp(i\theta) + exp(i2\theta)]$$
$$\Rightarrow exp(i2\theta)[cos(2\theta) – isin(2\theta) + cos(\theta) – isin(\theta) + 1 + cos(\theta) + isin(\theta) + cos(2\theta) + isin(2\theta)]$$
$$\Rightarrow exp(i2\theta)[2cos(2\theta) + 2cos(\theta) + 1]$$
Now I'm not sure where to go, when I try expanding anything it seems to become more complicated. Any help would be appreciated.
Best Answer
HINT
Notice you are dealing with a geometric progression whose common ratio equals $e^{i\theta}$: \begin{align*} \sum_{k=0}^{n-1}e^{ik\theta} = \frac{1 - e^{in\theta}}{1 - e^{i\theta}} \end{align*}
Make $n = 5$ and simplify the resulting expression.