Algebra-21 (CAT-2005)

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?

(1)5   (2) 3   (3) 2   (4) 6

Solution follows here:

Solution:

Given, g(x + 1) + g(x - 1) = g(x)

But as it is, this formula is not convenient to go for further iterations. Here, for finding g(x), we need to depend on its next higher iteration g(x+1), which is not convenient to proceed. So a small manipulation is required,

g(x+1) = g(x)-g(x-1) => here, to find g(x+1), we need to depend on its next lower iteration g(x), which is quiet possible.

g(x+1) = g(x)-g(x-1)

g(x+2) = g(x+1)-g(x) = [g(x)-g(x-1)]-g(x) = -g(x-1)

g(x+3) = g(x+2)-g(x+1) = [-g(x-1)]-[g(x)-g(x-1)] = -g(x)

g(x+4) = g(x+3)-g(x+2) = -g(x)-[-g(x-1)] = g(x-1)-g(x)

g(x+5) = g(x+4)-g(x+3) = [g(x-1)-g(x)]-[-g(x)] = g(x-1)

g(x+6) = g(x+5)-g(x+4) = g(x-1)-[g(x-1)-g(x)] = g(x)   ------ yes, we got it here,

g(x+6) = g(x) => p = 6

Answer (4)