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Fibonacci and Lucas

Authors
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    Name
    Vu Hung
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Problem Statement

Define

F1=1, F2=1, Fn=Fn−1+Fn−2,F_1=1,\ F_2=1,\ F_n=F_{n-1}+F_{n-2}, L1=1, L2=3, Ln=Ln−1+Ln−2.L_1=1,\ L_2=3,\ L_n=L_{n-1}+L_{n-2}.

Show that

Ln=Fn−1+Fn+1.L_n=F_{n-1}+F_{n+1}.

Hints

Set An=Ln+FnA_n=L_n+F_n and Bn=Ln−FnB_n=L_n-F_n, then compare with Fibonacci terms.

Solutions

From initial terms and recurrence,

An=Ln+Fn=2Fn+1,Bn=Ln−Fn=2Fn−1.A_n=L_n+F_n=2F_{n+1},\qquad B_n=L_n-F_n=2F_{n-1}.

Adding gives

2Ln=2Fn+1+2Fn−1,2L_n=2F_{n+1}+2F_{n-1},

so

Ln=Fn−1+Fn+1.L_n=F_{n-1}+F_{n+1}.

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