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Find the period for the function

Q.)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:
g(x+1) + g(x-1) =g(x)
g(x+2) + g(x) =g(x+1)
Adding the above two, we get
g(x+2) + g(x-1) =0 ——(i)
g(x+3) + g(x)=0
g(x+4) + g(x+1)=0
g(x+5) + g(x+2)=0 ——(ii)
From equation (i) and (ii), we get
g(x+5)= g(x-1)
Putting x+1 in place of x, we get
=>g(x+6) = g(x)
Hence, the required value of p=6


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