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**TL;DR Summary:**Prove that a sum of trigonometric ratios is periodic but not not simple harmonic.

We need to prove that ##x = sin{\omega t} + sin{2\omega t} + sin{4\omega t}## where ##x## is the displacement from the equilibrium position at time ##t##.

I can see that each term is a SHM, but not sure if the sum of these terms would be a simple harmonic motion.

Also, I can see that the time period of third term is one-fourth that of first term and time period of second term is half that of first term.

But after the above analysis, I am confused.