Fibonacci Sequence The famous Fibonacci sequence {un} is defined recursively as u1=1 u2=1 un

Fibonacci Sequence The famous Fibonacci sequence {un} is...

Fibonacci Sequence The famous Fibonacci sequence $\left\{u_{n}\right\}$ is defined recursively as $$ u_{1}=1 \quad u_{2}=1 \quad u_{n+2}=u_{n}+u_{n+1} \quad n \geq 1 $$ (a) Write the first eight terms of the Fibonacci sequence. (b) Verify that the $n$ th term is given by $$ u_{n}=\frac{(1+\sqrt{5})^{n}-(1-\sqrt{5})^{n}}{2^{n} \sqrt{5}} $$ (Hint: Show that $u_{1}=1, u_{2}=1,$ and $\left.u_{n+2}=u_{n+1}+u_{n} .\right)$

Fibonacci Sequence The famous Fibonacci sequence $\left\{u_{n}\right\}$ is defined recursively as
$$
u_{1}=1 \quad u_{2}=1 \quad u_{n+2}=u_{n}+u_{n+1} \quad n \geq 1
$$
(a) Write the first eight terms of the Fibonacci sequence.
(b) Verify that the $n$ th term is given by
$$
u_{n}=\frac{(1+\sqrt{5})^{n}-(1-\sqrt{5})^{n}}{2^{n} \sqrt{5}}
$$
(Hint: Show that $u_{1}=1, u_{2}=1,$ and $\left.u_{n+2}=u_{n+1}+u_{n} .\right)$

Key Concepts

Fibonacci Sequence

The Fibonacci sequence is a series of numbers where each term is obtained by summing the previous two terms, starting typically with the first two terms as 1. It is a central example of a recursive sequence and appears in many natural and theoretical contexts.

Recurrence Relation

A recurrence relation defines a sequence by expressing each term as a function of one or more preceding terms. In the Fibonacci sequence, the relation is u(n+2) = u(n+1) + u(n), establishing how subsequent terms are derived from prior ones.

Closed-form Expression

A closed-form expression provides a formula that calculates any term of a sequence directly, without recursion. For the Fibonacci sequence, this is represented by Binet's Formula, which involves roots and exponentiation to yield the nth term explicitly.

Mathematical Induction

Mathematical induction is a method of proof used to establish the validity of a statement for all natural numbers. In the context of the Fibonacci sequence, it is used to prove that the closed-form expression (Binet's Formula) correctly produces the sequence by verifying the base cases and then showing that the recurrence holds.

Transcript

00:01 Hello, so the famous fibonacci sequence, sometimes referred to as the huma sondra sequence, actually.

00:09 Actually, huma sondra kind of did it first, but it's a very famous sequence right in math.

00:15 And it begins while 1 -1.

00:16 So the first first two terms are while u -s -sub 1 is 1, u -sub 2 is 1, so we have 1, 1, and then what we have u -sub -m -plus 2 is equal to u -n, and then u -1, and then equal to 1.

00:31 So we have one and then one.

00:33 And then we have, well, our next term is going to be, well, u sub 2 plus, right, u sub m plus 2 is equal to u sub n plus 1.

00:45 So we just have u sub 3 is equal to u sub 2, right, plus u sub 1, which is going to be just, well, 1 plus 1, which is going to be 2.

00:58 And then we have our next term is just going to be, well, one plus two, which is going to be three.

01:05 And then our next term after that is just two plus three, which is five.

01:10 And our next term after that is three plus five.

01:13 Again, u sub four plus u sub five, which gives us u sub six, which is three plus five, which is eight.

01:21 And then we have, for n equals five, we get u sub five plus two.

01:26 So u sub 7 is equal to u sub 5 plus u sub 6.

01:29 So that's 3 plus 8, which is 11.

01:33 And then we have 8 plus 11, which is 19.

01:36 So the first eight terms are going to be 1.

01:40 1, 2, 3, 5, 8, 11, 19, and we can keep on going if we would like to...

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