# Question #56258

Feb 20, 2018

15

#### Explanation:

The sine function has a period of $2 \pi$ - that is, it repeats itself (for the first time) when the argument changes by $2 \pi$

Thus $\sin \left(2 \pi x\right)$ will repeat after its argument $2 \pi x$ changes by $2 \pi$ - this means $x$ has to change by only 1. The period of this function, is thus, 1.

Similarly, it is easy to see that the periods of the second and third terms in the sum are 3 and 5, respectively.

So, the there terms repeat whenever $x$ changes by multiples of 1, 3 and 5, respectively.

The sum will repeat when each of them repeats together - for this, $x$ has to change by a common multiple of 1, 3 and 5.

The period of the sum is, then, the least common multiple of 1, 3 and 5 - that is, 15.