Reducing exponents with a common base when terms are added

Question

I have a series of terms as follows:

$$e^{6x\pi.0} + e^{6x\pi.2} + e^{6x\pi.4} + e^{6x\pi.6}$$

Obviously the first term is just 1 but is there a way to specify the terms in one single term or shorten it somehow other than just 1 + ...?

i is an unknown in the expression

Answer 1

If the ".k" is multiply by $k$ then:

$$S=1+e^{6x\pi 2}+e^{6x\pi 4}+ e^{6x\pi 6}$$

$$Se^{6x\pi 2}=e^{6x\pi 2}+e^{6x\pi 4}+e^{6x\pi 6}+ e^{6x\pi 8}$$

Substract the two expressions:

$$Se^{6x\pi 2}-S=(e^{6x\pi 2}+e^{6x\pi 4}+e^{6x\pi 6}+ e^{6x\pi 8})-(1+e^{6x\pi 2}+e^{6x\pi 4}+ e^{6x\pi 6})$$

The only terms which remains are $1$ and $e^{6x\pi 8}$: $$S(e^{6i\pi 2}-1)=e^{6i\pi 8}-1$$

$$S = \frac{e^{6i\pi 8}-1}{e^{6i\pi 2}-1}$$

Answer 2

Hint $\ $ For $\ z = e^{\,\large 12\,\pi\,x}\,$ it is $\ 1 + z + z^2 + z^3 = \dfrac{z^4-1}{z-1},\ $ a Geometric Series.