= Solution
For direct-lattice points $x_n=na$, a <lattice-periodic plane wave> must have the same phase at every translated lattice point:
$$
e^{iG(x+a)}=e^{iGx},\qquad e^{iGa}=1.
$$
Thus $Ga=2\pi m$, with integer $m$, and the <reciprocal lattice> is
$$
\boxed{G_m=\frac{2\pi m}{a},\qquad m\in\mathbb Z}.
$$
Its spacing is $2\pi/a$. These plane waves execute an integer number of complete cycles within a direct-lattice period, so their wavelength magnitudes are $a/|m|$ for $m\ne0$.
A general electronic Bloch wave need not itself be lattice-periodic: wavevectors differing by $G_m$ have the same translation phase. The <reciprocal lattice> labels that equivalence, while a large finite specimen with <periodic boundary conditions> has a much finer grid of allowed electronic wavevectors.
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