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Question 27

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?

Solution

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According to given condition we have , g(x+1) = -g(x-1) + g(x)

Putting x=x+1 we get g(x+2) = g(x+1) - g(x) = -g(x-1)

Putting x=x+2 we get g(x+3)=-g(x)

Similarly g(x+4)=-g(x+1), g(x+5)=-g(x+2)=-g(x+1) + g(x)  and g(x+6) = g(x+1)-g(x+2)=g(x).

So p=6.

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