Let g (x) be a function such that `g (x + 1) + g (x - 1) = g (x)` for every real x. Then for what value of p is the relation `g (x + p) = -g (x)` necessarily true for every real x?

Options

A. 5
B. 3
C. 2
D. 6

Answer

To solve the problem, we need to analyze the given functional equation and find the value of \( p \) such that \( g(x + p) = -g(x) \).

### Step-by-Step Solution:

1. **Understand the Functional Equation**:
We start with the equation:
\[ g(x + 1) + g(x - 1) = g(x) \]
This equation holds for every real \( x \).

2. **Substitute Values**:
We will substitute \( x \) with \( x + 1 \) in the original equation:
\[ g((x + 1) + 1) + g((x + 1) - 1) = g(x + 1) \]
This simplifies to:
\[ g(x + 2) + g(x) = g(x + 1) \]

3. **Substitute Again**:
Now, we substitute \( x \) with \( x + 2 \):
\[ g((x + 2) + 1) + g((x + 2) - 1) = g(x + 2) \]
This simplifies to:
\[ g(x + 3) + g(x + 1) = g(x + 2) \]

4. **Relate the Equations**:
Now we have two equations:
- From step 2: \( g(x + 2) + g(x) = g(x + 1) \)
- From step 3: \( g(x + 3) + g(x + 1) = g(x + 2) \)

We can substitute \( g(x + 2) \) from the first equation into the second:
\[ g(x + 3) + g(x + 1) = g(x + 1) - g(x) \]
Simplifying this gives:
\[ g(x + 3) = -g(x) \]

5. **Identify the Value of \( p \)**:
We have found that:
\[ g(x + 3) = -g(x) \]
This means that if we let \( p = 3 \), then:
\[ g(x + p) = -g(x) \]
Therefore, the value of \( p \) is:
\[ p = 3 \]

### Final Answer:
The value of \( p \) for which \( g(x + p) = -g(x) \) is:
\[ \boxed{3} \]