Past exam of the mathematics course of the University of Cambridge / 2019 / ia / Paper 4 / 5E / b

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- Solution b

Solution

We prove both identities by mathematical induction. They hold at

n = 1

. Assume they hold at

n

. Using

F_{n - 1} = F_{n + 1} - F_{n}

,

F_{2 n + 2} = F_{2 n + 1} + F_{2 n} = F_{n}^{2} + F_{n + 1}^{2} + F_{n} (F_{n - 1} + F_{n + 1}) = F_{n + 1} (F_{n} + F_{n + 2}) .

Adding this to

F_{2 n + 1} = F_{n}^{2} + F_{n + 1}^{2}

and using

F_{n + 2} = F_{n} + F_{n + 1}

gives

F_{2 n + 3} = F_{n + 1}^{2} + F_{n + 2}^{2} .

These are precisely the two assertions with

n

replaced by

n + 1

, completing the induction.

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