Past exam of the mathematics course of the University of Cambridge / 2024 / ia / Paper 1 / 11E / b / Solution

Define the continuous functionAt the mesh points its values telescope:If one term is zero, the result follows immediately. Otherwise the terms cannot all have the same sign, so there are mesh points at which $h$ is positive and negative. The intermediate value theorem between those points gives an $x\in [0,(n-1)a]$ with $h(x)=0$. Thus the telescoping increment lemma yields

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