Find the norm of functional Consider the functional  from $l_2$.
$$
x=(x_n)\mapsto \sum \frac{x_n+x_{n+1}}{2^n}.
$$
What is  the norm of the functional?

It should not be very difficult to rewrite your functional to the form
$$x\mapsto \sum x_n y_n$$
where $y=y_n\in\ell_2$.
Then the norm of this functional is precisely $\|y\|_2$, i.e., it is the same as the $\ell_2$-norm of the sequence $y$.
To see this, just notice that you have the functional of the form
$$f(x)=\langle x,y \rangle,$$
where $\langle \cdot,\cdot \rangle$ is the standard inner product on $\ell_2$.
From Cauchy-Schwarz inequality you have
$$|f(x)| = |\langle x,y \rangle| \le \|y\|\cdot\|x\|,$$
i.e., $$\frac{|f(x)|}{\|x\|} \le \|y\|$$
which means that $\|f\|\le\|y\|$.
The opposite inequality follows from
$$f(y)=\langle y,y \rangle =\|y\|^2$$
or 
$$\frac{f(y)}{\|y\|}=\|y\|.$$
