An explicit formula for the n th term of the Fibonacci...

An explicit formula for the $n$ th term of the Fibonacci sequence is $$F_{n}=\frac{(1+\sqrt{5})^{n}-(1-\sqrt{5})^{n}}{2^{n} \sqrt{5}}$$ Apply algebra (not your calculator) to find the first two terms of this sequence and verify that these are indeed the first two terms of the Fibonacci sequence.

An explicit formula for the $n$ th term of the Fibonacci sequence is
$$F_{n}=\frac{(1+\sqrt{5})^{n}-(1-\sqrt{5})^{n}}{2^{n} \sqrt{5}}$$
Apply algebra (not your calculator) to find the first two terms of this sequence and verify that these are indeed the first two terms of the Fibonacci sequence.

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Key Concepts

Explicit Formulas for Recurrence Relations

Explicit formulas provide a method to compute terms of sequences defined by recurrence relations directly as functions of the index n. These formulas are useful for proving properties of sequences, analyzing their behavior, and verifying initial conditions without repetitive calculations.

Algebraic Manipulation and Verification

Algebraic manipulation involves substituting specific index values into an expression, simplifying radicals, and combining like terms to verify that a formula produces the correct sequence elements. This process confirms that an explicit formula matches the recurrence definition by checking the initial terms.

Fibonacci Sequence

The Fibonacci sequence is a famous example of a recursively defined sequence, where each term after the initial ones is the sum of its two immediate predecessors. It appears in various fields such as mathematics, computer science, and biology, and is notable for its relationship with the golden ratio.

Binet's Formula

Binet's Formula is an explicit closed-form expression that calculates the nth term of the Fibonacci sequence directly, bypassing the need to iterate through previous terms. It employs powers of irrational numbers related to the golden ratio and its conjugate to produce Fibonacci numbers exactly.

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