Let A be a C*-algebra and $\left\{\phi_n\right\}$ a weak* dense sequence in the state space. Putting $\phi=\sum_n 2^{-n} \phi_n$, can you show that $\phi$ is a state and the representation $\pi_\phi$ is faithful?
It's easy to show that $\phi$ is a state. But I don't know how to prove that $\pi_\phi$ is faithful.
We have $\phi(b)=\langle\xi,\pi_\phi(b)\xi\rangle$ for each $b\in\mathbf{A}$, where $\xi$ is the distinguished cyclic vector of norm $1$ in the space $H$ of the representation $\pi_\phi$. Take an arbitrary $a\in A$, $a\ne0$, and set $b=a^*a$. Since the sequence $\{\phi_n\}$ is weak* dense, it follows that $\phi_k(b)>0$ for some $k$ and hence $\phi(b)\ge 2^{-k}\phi_k(b)>0$. But then $\langle \pi_\phi(a)\xi,\pi_\phi(a)\xi\rangle=\phi(b)>0$, i.e., $\pi_\phi(a)\ne0$. QED