Question

If \(f(2)=5\) and \(f(x)\,f(x+1)=3\) for all real \(x\), the value of \(f(10)\) is ________________.

Hint:

A multiplicative relation \(f(x)f(x+1)=\text{const}\) often implies a 2-periodic solution: \(f(x+2)=f(x)\).

Answer 1

Correct Answer: 5

Step 1: From \(f(x)f(x+1)=3\) and \(f(x+1)f(x+2)=3\), dividing gives \(f(x+2)=f(x)\); hence \(f\) is period-2.
Step 2: Since \(f(2)=5\), all even integers share this value: \(f(4)=f(2)=5,\ldots,f(10)=5\). (Odd integers would be \(3/5\).)