CAT 2005 — QA Question 27
Periodic FunctionsEasy
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?
Answer & solution
Correct answer: 6
- A
5
- B
3
- C
2
6
Solution
g(x + 1) + g(x – 1) = g(x)
∴ g(x + 1) = g(x) – g(x – 1)
Now, let g(x − 1) = a and g(x) = b
∴ g(x + 1) = b – a
∴ g(x + 2) = b – a – b = –a
∴ g(x + 3) = −a – b + a = –b
∴ g(x + 4) = −b + a = a – b
∴ g(x + 5) = a – b + b = a = g(x – 1)
∴ g(x + 6) = a – a + b = b = g(x)
and so on.
Thus we observe that the values of g(x + 6) and g(x) are always equal.
Hence, option (d).