Fibonacci Sequence Let un=((1+√(5))^n-(1-√(5))^n)/(2^n √(5)) define the n th term of a sequence.

step 1 even u n is equal to 1 plus square root of 5 to the whole power n minus 1 minus square root of 5

Fibonacci Sequence Let un=((1+√(5))^n-(1-√(5))^n)/(2^n √(5))...

Fibonacci Sequence Let
$$
u_{n}=\frac{(1+\sqrt{5})^{n}-(1-\sqrt{5})^{n}}{2^{n} \sqrt{5}}
$$
define the $n$ th term of a sequence.
(a) Show that $u_{1}=1$ and $u_{2}=1$
(b) Show that $u_{n+2}=u_{n+1}+u_{n}$
(c) Draw the conclusion that $\left\{u_{n}\right\}$ is a Fibonacci sequence.

Key Concepts

Fibonacci Sequence

This is a sequence of numbers where each term after the first two is defined as the sum of the two preceding terms. It exemplifies how simple recursive rules can generate complex patterns and is widely recognized both in mathematical theory and real-world applications.

Recurrence Relation

A recurrence relation defines a sequence by specifying each term as a function of one or more of its previous terms. In the Fibonacci sequence, the key recurrence relation is that each term is the sum of the two prior terms, capturing the fundamental recursive structure of the sequence.

Binet's Formula

Binet's formula is the closed-form expression for finding the n-th term of the Fibonacci sequence. It uses irrational numbers, particularly elements derived from the square root of 5, to bypass iterative calculations and directly compute any term in the sequence.

Initial Conditions

Initial conditions refer to the specific starting values of a sequence that are necessary to uniquely determine all subsequent terms when a recurrence relation is defined. In the Fibonacci sequence, establishing the correct initial terms is critical to ensure the recursive process accurately generates the intended sequence.

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