本文探讨了卢卡斯数列与斐波那契数列之间的紧密联系，通过数学归纳法证明了两者之间的几个重要性质，并提供了一个包含前12项的数列表格供读者参考。
The Lucas numbers are closely related to the Fibonacci numbers and satisfy the same recursion relation Ln+1=Ln+Ln−1, but with starting values L1=1 and L2=3. The Lucas numbers are an example of a generalized Fibonacci sequence that satisfies fn+1=fn+fn−1 with starting values f1=p and f2=q.
(a) Using mathematical induction, prove that fn+2=Fnp+Fn+1q .
(b) Form a table containing the first 12 Fibonacci and Lucas numbers that can be referred to in the next two problems.
(c) Prove that Ln=Fn−1+Fn+1.
(d) Prove that Fn=1/5(Ln−1+Ln+1).
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