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4

Part of 2019 Dutch Mathematical Olympiad

Problems(1)

Fibonacci related sum a_i< 1 for a_n =\frac{1}{F_nF_{n+2}}

Source: Dutch NMO 2019 p4

1/9/2020
The sequence of Fibonacci numbers F0,F1,F2,...F_0, F_1, F_2, . . . is defined by F0=F1=1F_0 = F_1 = 1 and Fn+2=Fn+Fn+1F_{n+2} = F_n+F_{n+1} for all n>0n > 0. For example, we have F2=F0+F1=2,F3=F1+F2=3,F4=F2+F3=5F_2 = F_0 + F_1 = 2, F_3 = F_1 + F_2 = 3, F_4 = F_2 + F_3 = 5, and F5=F3+F4=8F_5 = F_3 + F_4 = 8. The sequence a0,a1,a2,...a_0, a_1, a_2, ... is defined by an=1FnFn+2a_n =\frac{1}{F_nF_{n+2}} for all n≥0n \ge 0. Prove that for all m≥0m \ge 0 we have: a0+a1+a2+...+am<1a_0 + a_1 + a_2 + ... + a_m < 1.
Fibonacci sequenceFibonacciSuminequalitiesalgebra