Let f : N → N be a function such that f(m + n) = f(m) + f(n) for every m, n∈N. If f(6) = 18, then f(2) . f(3) is equal to :

Solution

Maths sir · 3 steps

Break every number into 1s: additivity on naturals forces f(n) = n f(1).

1. Build up from 1. Put m = n = 1 to get f(2) = 2f(1). Then f(3) = f(2) + f(1) = 3f(1), and each later value adds one more f(1).

   f(n) = n f(1) for every n ∈ ℕ

2. Find f(1). f(6) = 6f(1) = 18, so f(1) = 3. That gives f(2) = 6 and f(3) = 9.
3. So. f(2) · f(3) = 6 × 9 = 54, which is option (b).

The idea to keep. If f(m+n) = f(m) + f(n) on natural numbers, then f(n) = n f(1). Find f(1) from the one given value, and every other value follows.

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