Fibonacci and Lucas

In the study of sequences and recurrence relations, the Fibonacci and Lucas numbers are classic companions that often hide neat algebraic connections. This topic fits squarely into the HSC Mathematics Extension 1 and 2 Sequences syllabus, and the following problem sharpens your ability to spot identities by manipulating related sequences. You'll learn how introducing simple auxiliary sequences can turn a seemingly tricky proof into a one‑line calculation.

Problem Statement

Define

$$F_1=1,\ F_2=1,\ F_n=F_{n-1}+F_{n-2},$$ $$L_1=1,\ L_2=3,\ L_n=L_{n-1}+L_{n-2}.$$

Show that

$$L_n=F_{n-1}+F_{n+1}.$$

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Hints

Set $A_n=L_n+F_n$ and $B_n=L_n-F_n$, then compare with Fibonacci terms.

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Solutions

We begin by building two new sequences that mix Lucas and Fibonacci numbers—exactly the hint’s suggestion. The idea is that each of these hybrids turns out to be a scaled version of a simple Fibonacci term, which makes the algebra effortless.

From initial terms and recurrence,

$$A_n=L_n+F_n=2F_{n+1},\qquad B_n=L_n-F_n=2F_{n-1}.$$

These equalities are justified because both $A_n$ and $B_n$ satisfy the same second‑order linear recurrence as the Fibonacci numbers, with initial values that are exactly twice the corresponding Fibonacci terms. Now we add the two equations to eliminate the $F_n$ term and recover $L_n$:

Adding gives

$$2L_n=2F_{n+1}+2F_{n-1},$$

so

$$L_n=F_{n-1}+F_{n+1}.$$

This cleanly proves that every Lucas number is simply the sum of its two neighbouring Fibonacci numbers.

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Takeaways

• The identity $L_n = F_{n-1} + F_{n+1}$ is a fundamental bridge between Lucas and Fibonacci sequences, showing they are more deeply intertwined than their different initial values suggest.
• Auxiliary sequences (like sums and differences) are a powerful tool for proving identities involving multiple recurrences—they can turn a messy derivation into a direct comparison of terms.
• Recognising that $A_n$ and $B_n$ behave like scaled Fibonacci numbers relies on checking initial values and confirming the recurrence matches; always verify a few early terms to build confidence.

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Further Readings

HSC Probability, HSC Last Resorts, HSC Sequences, HSC Polynomials