Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 318 / 3 / c

Table of contents
- Solution c

Solution

Induct on

k

, using the recursive divided-difference formula. After substituting the two induction hypotheses, use

f [t_{1}, \dots, t_{m}] - f [t_{0}, \dots, t_{m - 1}] = (t_{m} - t_{0}) f [t_{0}, \dots, t_{m}]

and the analogous identity for

g

; adjacent terms telescope. This proves

(f g) [t_{0}, \dots, t_{k}] = m = 0 \sum k f [t_{0}, \dots, t_{m}] g [t_{m}, \dots, t_{k}] .

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