= Solution
Let $C$ be the island's total electrostatic <capacitance> and $Q_{\rm ext}$ the charge offset induced by gates, reservoir potentials and fixed background charges. In the constant-capacitance model,
$$
U_N=\frac{(Ne-Q_{\rm ext})^2}{2C}+\text{constant independent of }N.
$$
Taking the difference gives
$$
\begin{aligned}
U_{N+1}-U_N
&=\frac{[(N+1)e-Q_{\rm ext}]^2-[Ne-Q_{\rm ext}]^2}{2C}\\
&=\boxed{\frac eC[(N+\tfrac12)e-Q_{\rm ext}]}.
\end{aligned}
$$
Changing gate voltage changes $Q_{\rm ext}$ approximately linearly, for example through $C_gV_g$.
For fixed $N$, $U_N(Q_{\rm ext})$ is a parabola. At small <temperature>/bias, the preferred integer is the one nearest $Q_{\rm ext}/e$; the equilibrium <energy> follows the lower envelope of these parabolas. Adjacent charge states become degenerate at \b[$Q_{\rm ext}=(N+\tfrac12)e$], allowing small-bias sequential current when the two barriers are transmissive.
The excess-electron charge $-Ne$ is a staircase. If “net charge” includes the induced compensating charge, define $q_{\rm net}=Q_{\rm ext}-Ne$; that residual charge is a sawtooth between $-e/2$ and $e/2$ in equilibrium. The opposite sign convention reverses the sawtooth. \b[State the charge convention:] an integer electron-charge staircase and a residual electrostatic-charge sawtooth describe different quantities.
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