Past exam of the mathematics course of the University of Cambridge / 2023 / ia / Paper 1 / 11E / b

The ordinary mean value theorem gives ${\Delta }_{h}f(a)=h{f}^{\prime }(b_1)$ for some $b_1\in (a,a+h)$. More generally, applying it to the difference of two translated functions shows inductively thatfor some $b_k\in (a,a+kh)$. Taking $k=n$ proves the result.