= Solution
The crystal has <periodic potential> $V(x+a)=V(x)$. By <Bloch's theorem>, an <energy eigenstate> may be chosen as
$$
\psi_k(x)=e^{ikx}u_k(x),\qquad u_k(x+a)=u_k(x).
$$
Expand the periodic factor in a <Fourier series>:
$$
u_k(x)=\sum_{n\in\mathbb Z}C_{k,n}e^{inG_1x},\qquad G_1=\frac{2\pi}{a}.
$$
Multiplication by $e^{ikx}$ gives the stated plane-wave expansion. Here $x$ is position, $a$ is the <primitive unit cell> length, $G_1$ is the primitive <reciprocal lattice> <wavenumber>, $n$ runs over every integer, and $C_{k,n}$ are <complex numbers> specifying the component <wave amplitudes>. The <wavevector> $k$ labels the <Bloch state> modulo $G_1$; $\hbar k$ is its <crystal momentum>, rather than generally its measured free-space <momentum>.
The component $e^{i(k+nG_1)x}$ is a <plane wave> of <wavevector> $k+nG_1$. All these components acquire the same <phase factor> $e^{ika}$ under $x\mapsto x+a$, since $e^{inG_1a}=1$. Their mixing therefore respects the crystal's translation symmetry. The coefficients and <energy eigenvalue> follow from the <Schrödinger equation>; the expression is not a claim that the coefficients are equal or that only finitely many are nonzero. The <energy band> index is suppressed in the notation, so different <energy bands> at a common $k$ generally have different coefficient sets.
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