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If f(x) is a periodic function with period p, show that it is

also periodic with period np for n = 2,3,4...

User Bijal
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Final answer:

A periodic function with period p is also periodic with period np for any positive integer n. This is because the function's value at any point x and x + np is the same, satisfying the definition of periodicity. This concept is related to the frequency of periodic motion in physics, where frequency is the inverse of the period.

Step-by-step explanation:

If f(x) is a periodic function with period p, then for any integer n = 2,3,4..., the function will also be periodic with period np. This can be shown by considering the definition of a period: a function f(x) is said to be periodic with period p if f(x + p) = f(x) for all x within the domain of f. Therefore, for any positive integer n, applying this definition repeatedly gives f(x + np) = f((x + (n-1)p) + p) = f(x + (n-1)p) = ... = f(x + 2p) = f(x + p) = f(x).

This demonstrates that the functional value at x and x + np is the same, hence making the function periodic with period np as well as with period p. This principle also connects with the concepts in physics where the frequency of a periodic motion is the reciprocal of the period T, where f = 1/T. Thus, if a motion has a period T, it also repeats every nT, where n is an integer.

User Artak
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