If a function satisfies the equation f(x + 1) + f(x –1) = 
f(x) ∀ x ∈ R. Prove that f(x) is periodic and also find the period.
Text Solution
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Sol. f(x + 1) + f(x – 1) = 
f(x)
replacing x by x + 1 and x –1 respectively, we get
f (x + 2) + f(x) = 
f(x + 1) …(i)
f(x) + f(x – 2) = 
f(x – 1) … (ii)
Adding (i) and (ii), we have
f(x + 2) + f(x–2) + 2f(x) = 
[f(x + 1) + f(x – 1)] = 3f(x)
⇒ f(x + 2) + f(x – 2) = f(x) … (iii)
replacing x by x + 2, we get
f(x + 4) + f(x) = f(x + 2) … (iv)
Adding (iii) and (iv)
f(x + 2) + f(x – 2) + f(x + 4) + f(x) = f(x) + f(x + 2)
⇒ f(x + 4) + f(x –2) = 0
replacing x by x + 2, we get
f(x + 6) + f(x) = 0
⇒ f(x + 6) = –f(x) … (v)
replacing x by x + 6, we get
f(x + 12) = –f(x + 6)
from (v) we have
f(x + 12) = f(x)
⇒ f(x) is periodic with period 12.
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