Question

Let $g(x)$ be a function such that $g(x+1)+g(x-1) = g(x)$ for all real $x$. For what $p$ does $g(x+p) = g(x)$ hold for all $x$?

• A. 5
• B. 3
• C. 2
• D. 6

Hint:

Solve functional equations by trying exponential forms and finding periods.

Answer 1

The Correct Option is B

1. Given the functional equation: \( g(x+1) + g(x-1) = g(x) \).
2. To find \( p \) such that \( g(x+p) = g(x) \) for all \( x \), we explore the periodicity and properties of this functional equation.
3. Assume a form of the sequence \( a_n = g(x+n) \). Then the equation becomes: \( a_{n+1} + a_{n-1} = a_n \).
4. This is a second-order linear homogeneous recurrence relation: \( a_{n+1} = a_n - a_{n-1} \).
5. Consider initial conditions \( a_0 = g(x) \) and \( a_1 = g(x+1) \). Compute further terms: \( a_2 = g(x+2) = a_1 - a_0 \), \( a_3 = g(x+3) = a_2 - a_1 = (a_1 - a_0) - a_1 = -a_0 \). 
6. Now, \( a_4 = g(x+4) = a_3 - a_2 = -a_0 - (a_1 - a_0) = -a_1 \).
7. Continuing, \( a_5 = g(x+5) = a_4 - a_3 = -a_1 - (-a_0) = a_0 - a_1 \).
8. Finally, \( a_6 = g(x+6) = a_5 - a_4 = (a_0 - a_1) - (-a_1) = a_0 \).
9. We see \( g(x+6) = g(x) \), indicating a potential periodicity.
10. Following the cyclic pattern: \( g(x+3) = -g(x) \), \( g(x+6) = g(x) \). The smallest positive integer satisfying \( g(x+p) = g(x) \) is \( p = 3 \).