Past exam of the mathematics course of the University of Cambridge / 2021 / ia / Paper 4 / 2E / ii

Solution

The stated formula is Cassini's identity. It holds for

n = 1

, and the recurrence gives

F_{n + 3} F_{n + 1} - F_{n + 2}^{2} = F_{n + 1}^{2} - F_{n + 2} F_{n},

which is the negative of the expression at index

n

. This proves the identity by induction.

Consequently

a_{n + 1} - a_{n} = \frac{F _{n + 2} F _{n} - F _{n + 1}^{2}}{F _{n + 1} F _{n}} = \frac{( - 1 ) ^{n + 1}}{F _{n + 1} F _{n}} .

The positive Fibonacci numbers tend to infinity, so this difference tends to zero.

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