Question

Let a function f saisfy f(x + 1) = f(x) + x, ∀ x ∈ N where f(1) = 0. Find f(x).

Answer 1

Given f(1) = 0 

f(2) = f(1) + 1 = 0 + 1 = 1 

f(3) = f(2) + 1 = 1 + 1 = 2 

f(4) = f(3) + 1 = 2 + 1 = 3, ... 

Now, f(2) – f(1) = 1 

f(3) – f(2) = 2 

f(4) – f(3) = 3 ... ... ... ... ... ... ... ... ... f(x + 1) – f(x) = x 

On addition, we get, 

f(x + 1) – f(1) = 1 + 2 + 3 + ... + x 

⇒ f(x + 1) = f(1) + 1 + 2 + 3 + ... + x 

⇒ f(x + 1) = 0 + 1 + 2 + 3 + ... + x = x(x - 1)/2

Replacing x by x – 1, we get,

f(x) = ((x - 1)(x - 2))/2 for all x in N.