Question

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?

• A. 5
• B. 3
• C. 2
• D. 6

Answer 1

The correct option is D 6

g(x + 1) + g(x - 1) = g(x)
$\Rightarrow g(x+1)=g\left(x\right)-g(x-1)$
Using x = x + 5
$\Rightarrow g(x+6)=g(x+5)-g(x+4)$
$=g(x+4)-g(x+3)-g(x+4)=-g(x+3)$
$=-\left[g\right(x+2)-g(x+1\left)\right]$
$=-g(x+1)+g\left(x\right)+g(x+1)=g\left(x\right)$
Hence, p = 6.