Find the Twenty-fourth Harmonic

The questions below are due on Thursday February 27, 2025; 02:00:00 PM.
 
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Let x[n] represent the sum of three signals:

x[n]=x_1[n]+x_2[n]+x_3[n]

where x_1[n] is periodic with a period of 3,

x_2[n] is periodic with a period of 4,

and x_3[n] is periodic with a period of 5,

Let X[k] represent the Fourier series coefficients for x[n]:

X[k] = {1\over N}\sum_{n=\left\langle N\right\rangle}x[n]e^{-j{2\pi k n \over N}}

where N represents the smallest positive integer for which x[n]=x[n+N] for all n.

Determine a closed-form expression for X[24], and explain your process.

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