Final answer:
The proof of the eigenvalues for the Fibonacci sequence uses irrational numbers, and when calculating large values, approximations are necessary due to limitations in representing these numbers exactly in decimal form. Furthermore, rounding errors in floating-point arithmetic for large n contribute to the result being an estimate rather than an exact value.
Step-by-step explanation:
The reason the proof of the eigenvalues of the Fibonacci sequence doesn't produce exact values for the nth Fibonacci number lies in the nature of the Fibonacci sequence itself and the mathematical technique used to find its terms. The standard approach to finding the nth term of the Fibonacci sequence involves the use of the characteristic equation of the sequence's generating matrix, which yields two eigenvalues often denoted φ (the golden ratio) and ψ (φ being the positive root and ψ the negative root of the characteristic equation).
To calculate the nth term of the Fibonacci sequence, Binet's Formula is frequently used, which is an explicit form that involves both eigenvalues:
F(n) = (1/√5)[(φn - ψn)]
However, since φ and ψ are irrational numbers, when calculating large values of n, the term ψn can become very small and thus contribute only slightly to the final number. This leads to the result being an approximation rather than an exact value due to the difficulty in representing irrational numbers with complete accuracy in a finite decimal system. Furthermore, the floating-point arithmetic used in practical computations can introduce rounding errors, particularly for large values of n. Hence, the formula gives a very close approximation but not the exact value for large nth terms in the Fibonacci sequence.