The Lucas numbers l0, l1, l2, …, ln …are defined using the same recurrence relation defining the

The Lucas numbers l0, l1, l2, …, ln …are defined using the...

Key Concepts

Mathematical Induction

Mathematical induction is a proof technique used to verify statements for all natural numbers. In the context of sequences defined by recurrence relations, induction is crucial for proving identities and properties that hold for each term in the sequence.

Recurrence Relations

Recurrence relations are equations that express each term of a sequence as a function of preceding terms. They are fundamental in the study of sequences such as the Fibonacci and Lucas numbers, where the same recursive pattern is used to build all subsequent terms.

Fibonacci Sequence

The Fibonacci sequence is a classical sequence where each term is the sum of the two preceding terms, starting typically from 0 and 1. It serves as a cornerstone in understanding recursive processes and has numerous applications in mathematics, science, and art.

Lucas Sequence

The Lucas sequence shares the same recurrence relation as the Fibonacci sequence but begins with different initial conditions. Studying the Lucas sequence helps illuminate how changes in starting values can influence the properties and identities of sequences that obey the same recursive rules.

Sequence Identities

Sequence identities are formulas that express relationships between different terms within one or more sequences. They are essential in revealing deeper connections between sequences, such as the relationships between Fibonacci and Lucas numbers, and often involve algebraic manipulation of their defining recurrences.

Transcript

00:02 We are given the definition of the lucas numbers and initial conditions for the lucas numbers.

00:09 We are asked to answer some questions about them.

00:12 So they satisfy the recurrence relation ln equals ln minus 1 plus ln minus 2, and the initial conditions l0 equals 2 and l1 equals 1.

00:25 In part a, we're asked to write the lucas number ln in terms of the nth, in terms of the n minus 1st and n plus first, fibonacci numbers.

00:48 So in this case, we'll use induction.

00:55 So if n is equal to 2, we have that l2, this is equal to from the definition of lucas numbers, l1 plus l0, which is 3, and we have that f1 plus f3, this is going to be 1 plus 2, which is also 3, so that it works for the base case and for the inductive step we're going to suppose that the lucas number ln is equal to fn minus 1 plus fn plus 1 for all n less than or equal to k for some k which is going to be greater than or equal to 2 and this is going to be all n between 2 and k so we have that l sub k plus 1 by the recurrence relation is lk plus lk minus 1.

02:54 By the inductive hypothesis, this is equal to fk minus 1 plus fk plus 1, which is equal to or plus and this is fk minus 2 plus fk.

03:23 And this can be rewritten as fk minus 2 plus fk minus 1 plus fk plus fk plus 1 and we have by the recurrence relation for the fibinacci numbers that this is equal to fk plus fk plus 2 and this is what we wanted to show so it follows that l n is equal to fn minus 1 plus fn plus 1 for all natural numbers greater than are equal to 2 by induction.

04:26 And in part b, we're asked to find an explicit formula for the lucas numbers.

04:37 So we have the lucas numbers have a linear homogenous recurrence relation.

04:42 And so the characteristic equation is r squared minus r minus 1 equals 0, which has roots r equals 1 plus or minus the square root of 1 plus 4, which is 5, all over 2...

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