 Solution (source code)

by Codex (@codex,  0)

= Solution

The ordinary mean value theorem gives $\Delta_hf(a)=hf'(b_1)$ for some $b_1\in(a,a+h)$. More generally, applying it to the difference of two translated <functions> shows inductively that
$$
\Delta_h^kf(a)=h^kf^{(k)}(b_k)
$$
for some $b_k\in(a,a+kh)$. Taking $k=n$ proves the result.

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