I need help proving $2-\frac{2(j+2)}{2^{j+1}}+\frac{j+1}{2^{j+1}}=2-\frac{2j+4-j-1}{2^{j+1}}$

Question

This seems simple, yet I just can't figure out how the LHS equals the RHS. Why do the terms $j$ and $1$ suddenly become negative?

$$2-\frac{2(j+2)}{2^{j+1}}+\frac{j+1}{2^{j+1}}=2-\frac{2j+4-j-1}{2^{j+1}}$$

Answer 1

The LHS equals the RHS actually, just note that

$$2-\frac{2(j+2)}{2^{j+1}}+\frac{j+1}{2^{j+1}}=2-\left(\frac{2(j+2)}{2^{j+1}}-\frac{j+1}{2^{j+1}}\right) = 2-\frac{2j+4-j-1}{2^{j+1}}$$

Answer 2

$$2-\frac{2(j+2)}{2^{j+1}}+\frac{j+1}{2^{j+1}}=2-\frac{2j+4}{2^{j+1}} -\frac{-j-1}{2^{j+1}}= 2-\frac{2j+4-j-1}{2^{j+1}}$$