Distributing 2 times k equals k?

Question

in my current discrete mathmatics course I have this calculation at the end of a proof:

$$\frac {k(k+1)+2(k+1)}2=\frac {(k+1)(k+2)}2$$

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I dont under stand why 2*k ends up being k? I guessing there is some calculations being abstracted away, can someone please exaplain?

Answer 1

Just factor out the $(k+1)$ term

or if you prefer expand and then factor:

$$\frac {k(k+1)+2(k+1)}2=\frac {k^2+3k+2}2=\frac {(k+1)(k+2)}2$$

Answer 2

With colors: $$\frac{\color{blue}{k}\color{red}{(k+1)}+\color{green}{2}\color{red}{(k+1)}}{2} =\frac{(\color{blue}{k}+\color{green}{2})\color{red}{(k+1)}}{2}$$ The red part $\color{red}{(k+1)}$ is seen in both terms of the left-hand-side, so can be "moved out" to the right.