I'm trying to understand this proof that
\begin{equation}
\int_{\mathbb{R}^n}\frac{|f(x)-f_{Q_1}|}{1+|x|^{n+1}}dx\leq A\|f\|_{BMO}
\end{equation}
where
$$\|f\|_{BMO}=\sup_{Q\text{ is a cube containing x}}\left\{\frac{1}{|Q|}\int_Q|f(x)-f_Q|dx\right\}$$
and $f_Q=\frac{1}{|Q|}\int_Qf(x)dx$, for any cube $Q$ in $\mathbb{R}^n$, and $Q_1$ is the cube with side lenght $1$ centered at $0$. The proof goes as follows:
Let $Q_{2^k}$ be the cube with center $0$ and side lengths $2^k$, for each $k\in\mathbb{N}_0$. Then of course $|\int_{Q_{2^{k-1}}}[f(x)-f_{Q_{2^{k}}}]dx|\leq2^{nk}\|f\|_{BMO}$ and therefore $|f_{Q_2^{k-1}}-f_{Q_{2^k}}|\leq 2^n\|f\|_{BMO}$. Adding these inequalities gives $|f_{Q_{2^k}}-f_{Q_1}|\leq 2^nk\|f\|_{BMO}$, and so
$$ \int_{Q_{2^k}}|f(x)-f_{Q_1}|dx\leq 2^{nk}[1+2^nk]\|f\|_{BMO}$$
What I do not understand is this last step: after obtaining the previous inequality, the book simply says:
"A last addition in $k$ gives the result."
I have tried taking limits, supremums, summing the cubic shells, etc... but I can't manage to obtain the result.