how can (k+1)! (k+2)-1 be (k+2)!-1?

Question

how can $(k+1)! (k+2)-1$ be $(k+2)!-1$ ? Can someone explain how it works?

I already tried to expand it but didn't work.

Answer 1

Use the definition of factorial.

Since $(k+2)! = (k+2) \cdot (k+1) \cdot k \cdots 2 \cdot 1 = (k+2) \cdot (k+1)!$, you have

$$ (k+1)! (k+2) - 1 = (k+2)! - 1.$$

Unless I misunderstood your question?