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Assume constant time spacing: \( \Delta t = t_{n+1}-t_n=\mbox{const} \) $$ \begin{align*} u_0 &= I,\\ u_1 & = u^0 - a\Delta t u^0 = I(1-a\Delta t),\\ u_2 & = I(1-a\Delta t)^2,\\ u^3 &= I(1-a\Delta t)^3,\\ &\vdots\\ u^{N_t} &= I(1-a\Delta t)^{N_t} \end{align*} $$
Ooops - we can find the numerical solution by hand (in this simple example)! No need for a computer (yet)...
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