Suppose f is a function satisfying f(x + y) = f(x) + f(y) for all x,y ∈ N and f(1) = 1/5. If Σₙ₌₁ᵐ f(n)/n(n + 1)(n + 2) = 1/12, then m is equal to __________.

Solution

Maths sir · 4 steps

Cauchy's equation on naturals makes f(n) = n/5, then cancel the n and let the sum telescope.

1. Find f(n). Put y = 1 to get f(x+1) = f(x) + 1/5. So f goes up by 1/5 at each step, and f(n) = n/5 for every natural n.
2. Cancel the n. The general term becomes (n/5)/n(n+1)(n+2) = 1/5 · 1/(n+1)(n+2). Split it as 1/(n+1)(n+2) = 1/(n+1) - 1/(n+2).

   Σₙ₌₁ᵐ f(n)/n(n+1)(n+2) = 1/5(1/2 - 1/(m+2))

3. Set it equal. 1/5(1/2 - 1/(m+2)) = 1/12 gives 1/2 - 1/(m+2) = 5/12. So 1/(m+2) = 1/12, and m + 2 = 12.
4. So. m = 12 - 2 = 10.

The idea to keep. When f(x+y) = f(x) + f(y) holds on naturals, f(n) = n f(1). After that, look for a factor that cancels and a difference of fractions that telescopes.

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