The number of functions
f:{ 1,2,3,4} → { a ∈ Z|a| ≤ 8}
satisfying f(n) + 1/nf(n + 1) = 1,∀ n ∈ { 1,2,3} is

Solution

Maths sir · 4 steps

Put everything in terms of f(1), then let the tight bound on f(4) decide how many values survive.

1. Rewrite the rule. Multiply the condition by n to get f(n+1) = n(1 - f(n)). So once f(1) is chosen, every other value is fixed.
2. Chain from f(1). Let f(1) = x. Then f(2) = 1 - x, f(3) = 2(1 - (1-x)) = 2x, and f(4) = 3(1 - 2x).

   f(1)=x, f(2)=1-x, f(3)=2x, f(4)=3-6x

3. Apply the bounds. Every value must be an integer from -8 to 8, and x itself is an integer. |2x| ≤ 8 gives -4 ≤ x ≤ 4. |3-6x| ≤ 8 gives -5 ≤ 6x ≤ 11, so x = 0 or x = 1. Both also satisfy |x| ≤ 8 and |1-x| ≤ 8.

   x = f(1)	f(2)	f(3)	f(4)
   0	1	0	3
   1	0	2	-3

4. So. Exactly two choices of f(1) keep all four values inside the codomain, so there are 2 such functions, option (a).

The idea to keep. When a functional equation links f(n) to f(n+1), write every value in terms of the first one. The value with the largest coefficient gives the tightest bound.

Answer from the NTA final answer key. Solution by PrepWiser.