Sequence - Part 9

This is the last post of our series; here is the link to "Sequence - Part 1" if you haven't read it. Today we will do some more exercises on sequence. We will prove some interesting identities. For the Pell sequence $$P_0=0, ~~P_1 = 1, ~~P_n = 2 P_{n-1} + P_{n-2},$$ and the companion Pell sequence $$H_0=1, ~~H_1 = 1, ~~H_n = 2 H_{n-1} + H_{n-2},$$ we will show that $$H_n^2 - 2 P_n^2 = (-1)^n.$$
For the Fibonacci sequence $$F_0 = 0, ~~F_1 = 1, ~~F_n = F_{n-1} + F_{n-2},$$ we will prove the following identity $$\frac{F_{2013(n+1)} - F_{2013 (n−1)}}{F_{2013 n}} = \frac{F_{2013(n^{2013}+1)} - F_{2013 (n^{2013}−1)}}{F_{2013 n^{2013}}}.$$

Sequence - Part 8

In the last post, we learn how to determine a trigonometric formula for a sequence in the case when the characteristic equation has complex roots. Today we will solve more exercises for this case.

Sequence - Part 7

This is the 7th part of our series on sequences. Today we will learn how to solve a linear recurrence equation in the case when the characteristic equation has complex roots. In this case, we can express the complex roots of the characteristic equation in trigonometric form and then use de Moivre's identity to obtain a trigonometric formula for the sequence.

Sequence - Part 6

Today we will learn about difference operator and use it to prove a fundamental theorem for linear recurrence equations.

A fundamental theorem for linear recurrence equations. Suppose that the characteristic equation can be factored as $$f(x) = a_k x^k + a_{k-1} x^{k-1} + \dots + a_0 = (x - z)^j (b_s x^s + b_{s-1} x^{s-1} + \dots + b_0)$$ and $$f_n = p(n)~z^n,$$ where $p(n)$ is a polynomial of degree less than $j$. Then the sequence $f_n$ satisfies the recurrence equation 
$$a_k f_{n} + a_{k-1} f_{n-1} + \dots + a_1 f_{n-k+1} + a_0 f_{n-k} = 0.$$

Sequence - Part 5

Today we will go through some examples. The first type of examples involves solving a linear recurrence equation to derive a general formula for a sequence. The second type of examples does the reverse process, in which, we are given a formula of a sequence and asked to determine its recurrence equation.

Sequence - Part 4

Today we will learn how to solve a linear recurrence equation and derive a general formula for a sequence. This method will work for all cases, including the case when the characteristic equation has multiple roots.  

Sequence - Part 3

This is the third part of our series on sequences. There will be no new material in this post. The main purpose of this post is to go through some examples so we can get used to the method of solving recurrence equations. If you have not read the previous parts of our series, please read them here: Part 1, Part 2.

Sequence - Part 2

This is the second part of our series on sequences. Please have a careful read of Part 1 before you continue to this post. Today, we will introduce some more terminologies about sequences and we will present a general method to solve a linear recurrence equation.

Sequence - Part 1

Today we will begin our first post of a series on recurrence sequences. The purpose of this series is to show you how to derive a general formula for a sequence that satisfies a linear recurrence equation. We start with some basic operations on sequences. We will learn about addition of sequences and multiplication of a sequence with a number.