Binet Formula

This formula gives the nth Fibonacci number directly, without generating all earlier terms. It comes from solving the Fibonacci recurrence F_n = F_{n-1} + F_{n-2} like a linear difference equation: the sequence is a combination of two exponential terms, and the constants are chosen to match the initial values. The term involving \psi is very small in magnitude for large n, which is why the answer is always an integer after the two terms are combined.

Use this when you want an explicit expression for a Fibonacci number, especially in proofs, algorithm analysis, or when connecting Fibonacci numbers to algebra and eigenvalues. It is useful for estimating growth or deriving identities, even though for exact computation with large n one usually prefers integer recurrence or fast doubling methods.

• Using \psi = -1/\varphi incorrectly as 1-\varphi without checking the sign
• Rounding \varphi^n too early, which can give the wrong integer after division by \sqrt{5}
• Forgetting that the formula returns exact Fibonacci numbers only when the correct indexing convention for F_0 and F_1 is being used