Assertion: A combination of two simple harmonic motions with a arbitrary amplitudes and phases is not necessarily periodic.
Reason: A periodic motion can always be expressed as a sum of infinite number of harmonic motions with appropriate amplitudes.

Options

A. If both assertion and reson are true and reason is the correct explanation of assertion
B. If both assertion and reason are true but reason is not the correct explanation of assertion.
C. If assertion is true but reason is false.
D. If both assertion and reason are false.

Explanation

To solve the question, we need to analyze both the assertion and the reason provided.

### Step 1: Understand the Assertion
The assertion states that a combination of two simple harmonic motions (SHMs) with arbitrary amplitudes and phases is not necessarily periodic.

**Explanation**:
- For two SHMs to be periodic when combined, their frequencies must have a specific relationship. Specifically, if the frequency of one SHM is an integral multiple of the other, the combination will be periodic. If not, the combination may not repeat itself after a fixed interval, hence not periodic.

### Step 2: Understand the Reason
The reason states that a periodic motion can always be expressed as a sum of an infinite number of harmonic motions with appropriate amplitudes.

**Explanation**:
- This statement is true in the context of Fourier series, where any periodic function can be represented as a sum of sine and cosine functions (harmonic motions). However, this does not directly explain why the assertion is true.

### Step 3: Determine the Truth of Each Statement
- The assertion is **true** because the combination of two SHMs is periodic only under certain conditions (integral multiple of frequencies).
- The reason is also **true**, but it does not provide a correct explanation for the assertion.

### Step 4: Conclusion
Since both the assertion and the reason are true, but the reason does not correctly explain the assertion, the correct answer is that both are true, but the reason is not a correct explanation of the assertion.

### Final Answer:
The correct option is B: Both assertion and reason are true, but the reason is not a correct explanation of the assertion.

---