# Complex arithmetic/Geo Series proof

1. Jan 27, 2010

### PhysicsMark

1. The problem statement, all variables and given/known data
Applications from multiple-slit diffraction involve sums like the following. Prove that:

$$\sum^{N-1}_{n=0}$$ cos (nx) = $$\frac{sin(N(x/2))}{sin(x/2)}$$ * cos((N-1)x/2)

2. Relevant equations

According to my instructions, this should involve only algebraic manipulations

Also there is this hint:

Use the geometric series formula:

$$\sum^{M}_{n=0}$$ r$$^{n}$$= $$\frac{1-r^{M+1}}{1-r}$$

3. The attempt at a solution
Once again, I apologize for the poor use of Latex. I hope the equation is easily understood. This problem comes from the complex arithmetic section of my book. I originally followed a lot of different trig identity paths until I re-read the part about using only algebraic manipulations.

If possible, I would like a small shove past the hint. I'm not asking for a complete proof here, only a little more light in the right direction. Thanks for taking the time to read.

2. Jan 27, 2010

### Dick

cos(nx) is the real part of exp(inx). So the sum of cos(nx) is the real part of the sum of exp(ix)^n. The exponential form is a geometric series.