GATE 2023 BT – Question 60
If $f(2)=5$ and $f(x)f(x+1)=3$ for all real x, the value of $f(10)$ is ___________.
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Correct answer: 5
Explanation
**Step 1: a recurrence.** From $f(x)\,f(x+1)=3$,
$$f(x+1)=\frac{3}{f(x)} .$$
**Step 2: apply it twice.**
$$f(x+2)=\frac{3}{f(x+1)}=\frac{3}{3/f(x)}=f(x).$$
So $f$ repeats with period 2.
**Step 3: value.** $10-2=8$ is a multiple of 2, so
$$f(10)=f(2)=\mathbf{5}.$$