Sets, Relations & Functions
Counting Functions / Integer-valued Functions
nta_pyq_2023_jan
Grade 11
Question:
The number of functions $f:\{1,2,3,4\} \to \{a \in \mathbb{Z}: |a| \le 8\}$ satisfying $f(n) + \dfrac{1}{n}f(n+1) = 1$, $\forall n \in \{1,2,3\}$ is
Step-by-Step Solution
Key Concept: $f(n+1)$ must be divisible by $n$; enumerate valid integer solutions for $f(4), f(3), f(2), f(1)$.
From the recurrence, $f(4) \in \{-3,3\}$ (odd multiples of 3 with $|f(4)|\le8$), giving 2 valid function chains.
Correct Answer: 4