SEQUENCE
* In mathematics, informally speaking, a sequence is an ordered list of objects (or events). Like a set, it contains members (also called elements, or terms). The number of ordered elements (possibly infinite) is called the length of the sequence. Unlike a set, order matters, and exactly the same elements can appear multiple times at different positions in the sequence. Most precisely, a sequence can be defined as a function whose domain is a countable totally ordered set, such as the natural numbers. * For example, {M, A, R, Y} is a sequence of letters with the letter 'M' first and 'Y' last. This sequence differs from {A, R, M, Y}. Also, the sequence {1, 1, 2, 3, 5, 8}, which contains the number 1 at two different positions, is a valid sequence. Sequences can be finite, as in this example, or infinite, such as the sequence of all even positive integers {2, 4, 6,...}. Finite sequences are sometimes known as strings or words and infinite sequences as streams. The empty sequence { } is included in most notions of sequence, but may be excluded depending on the context. ARITHMETIC SEQUENCE
* A sequence such as 1, 5, 9, 13, 17 or 12, 7, 2, –3, –8, –13, –18 which has a constant difference between terms. The first term is a1, the common difference is d, and the number of terms is n.
Explicit Formula: an = a1 + (n – 1)d
Example 1: 3, 7, 11, 15, 19 has a1 = 3, d = 4,
and n = 5. The explicit formula is
an = 3 + (n – 1)·4 = 4n – 1
Example 2: 3, –2, –7, –12 has a1 = 3, d = –5, and n = 4. The explicit formula is
an = 3 + (n – 1)(–5) = 8 – 5n
GEOMETRIC SEQUENCE
* Before talking about geometric sequence, in math, a sequence is a set of numbers that follow a pattern. We call each number in the sequence a term.
For examples, the following are sequences:
2, 4, 8, 16, 32, 64, .......
243, 81, 27, 9, 3, 1, ...............
A geometric sequence is a sequence where each term is found by multiplying or dividing the same...
...feeling which I experienced at the time of my initiation to this
extraordinarily beautiful world of ideas which we call higher mathematics.
CONFESSION OF THE READER. Recently our professor of
mathematics told us that we begin to study a new subject which
he called calculus. He said that this subject is a foundation of higher
mathematics and that it is going to be very difficult. We have already
studied real numbers, the real line, infinite numerical sequences, and
limits of sequences. The professor was indeed right saying that comprehension of the subject would present difficulties. I listen very
carefully to his explanations and during the same day study the
relevant pages of my textbook. I seem to understand everything, but
at the same time have a feeling of a certain dissatisfaction. It is difficult for me to construct a consistent picture out of the pieces obtained
in the classroom. It is equally difficult to remember exact wordings
and definitions, for example, the definition of the limit of sequence.
In other words, I fail to grasp something very important.
Perhaps, all things will become clearer in the future, but so far
calculus has not become an open book for me. Moreover, I do not
see any substantial difference between calculus and algebra. It seems
6
Preface
that everything has become rather difficult to perceive and even more
difficult to keep in my memory.
COMMENTS OF THE AUTHOR....
...Arithmetic Sequence is a sequence of numbers in which each succeeding term differs from the preceding term by the same amount. This amount is known as the common difference and can be found using a specific formula by substituting the numbers from the word problem into the equation. When you plug in all the information, you are able to find out the money that needs to be spent and saved in the following word problems.
35. A person hired a firm to build a CB radio tower. The firm charges $100 for labor for the first 10 feet. After that, the cost of the labor for each succeeding 10 feet is $125 more than the preceding 10 feet will cost $125, the next ten feet will cost $150 etc. How much will it cost to build a 90 foot tower?
an=a1+ (n1) d
a125=100+ (1251) (150)
a125=100+124(150)
a125=100+18600
a125=18700
sn =n (a1 + an) / 2
= 125 (100+18700) /2
=125(1880) /2
=62.5 (18800) =1175000
The cost to build a 90foot tower is $11,750.
37. A person deposited $500 in a savings account that pays 5% annual interest that is compound yearly. At the end of 10 years, how much money will be in the savings account?
S+ (0.5) S n=10
S+ (1+0.5) r=1.05
S (1.05) a1= 500(1.05) =525
an= a1(rn1)
a10=525(1.059)
a10=525(1.551328216)
a10=814.4473134
The balance in the savings account at the end of 10 years will be $814.44.
I chose to use the Arithmetic...
...Sequences and Convergence
Let x1 , x2 , ..., xn , ... denote an infinite sequence of elements of a metric space
(S, d). We use {xn }∞
n=1 (or simply {xn }) to denote such a sequence.
Definition 1 Consider x0 ∈ S. We say that the sequence {xn } converges to x0
when n tends to infinity iff: For all > 0, there exists N ∈ N such that for all
n > N , d(xn , x0 ) <
We denote this convergence by lim xn = x0 or simply xn −→ x0 .
n→∞
Example 2 Consider the sequence {xn } in R, defined by xn = n1 . Then xn −→
0.
The way to prove this is standard: fix > 0. We need to find N ∈ N such that
for all n > N , d(xn , 0) < . We have d(xn , 0) = xn − 0 =  n1 
So it is enough that n1 < , or equivalently n > 1 . So choosing N > 1 we know
that for all n > N , d(xn , 0) < .
The fact that we define the concept of convergence does not imply that every
sequence converges. This is illustrated in the next two examples. Let’s begin
with a remark about what it means for a sequence {xn } not to converge to x0 .
Remark: To know what the nonconvergence of a sequence means, we need
to write the negation of the definition of convergence. That reduces to: There
exists > 0, such that for all N ∈ N, there exists n > N such that d(xn , x0 ) ≥ .
For the ones of you familiar with propositional logic, notice that convergence to
x0 can be written as
(∀ > 0)(∃N ∈ N)(∀n > N )d(xn , x0 ) <
Its negation...
...Anatolia College 
Mathematics HL investigation

The Fibonacci sequence 
Christos Vassos

Introduction
In this investigation we are going to examine the Fibonacci sequence and investigate some of its aspects by forming conjectures and trying to prove them. Finally, we are going to reach a conclusion about the conjectures we have previously established.
Segment 1: The Fibonacci sequence
The Fibonaccisequence can be defined as the following recursive function:
Fn=un1+ un2
Where F0=0 and F1=1
Using the above we can find the first eight terms of the sequence. An example of calculations is given below:
F2=F1F0F2=1+0=1
We are able to calculate the rest of the terms the same way:
F0  F1  F2  F3  F4  F5  F6  F7 
0  1  1  2  3  5  8  13 
Segment 2: The Golden ratio
In order to define the golden ratio we need to examine the following sketch:
The line above is divided into two segments in such a way that ABAP=APPB
The ratio described above is called the golden ratio.
If we assume that AP=x units and PB=1 units we can derive the following expression:
x+1x=x1
By solving the equation x2x1=0 we find that: x=1+52
Segment 3: Conjecture of φn
In this segment we examine the following geometric sequence:
φ,φ2,φ3…
Since x=1+52 can simplify φ by replacing the value of x to the formula of the golden ratio we discussed...
...
This work MAT 126 Week 1 Assignment  Geometric and Arithmetic Sequence shows "Survey of Mathematical Methods" and contains solutions on the following problems:
First Problem: question 35 page 230
Second Problem: question 37 page 230
Mathematics  General Mathematics
Week One Written Assignment
Following completion of your readings, complete exercises 35 and 37 in the “Real World Applications” section on page 280 of Mathematics in Our World .
For each exercise, specify whether it involves an arithmetic sequence or a geometric sequence and use the proper formulas where applicable . Format your math work as shown in the Week One Assignment Guide and be concise in your reasoning. Plan the logic necessary to complete the exercise before you begin writing. For an example of the math required for this assignment, please review the Week One Assignment Guide .
The assignment must include ( a ) all math work required to answer the problems as well as ( b ) introduction and conclusion paragraphs.
Your introduction should include three to five sentences of general information about the topic at hand.
The body must contain a restatement of the problems and all math work, including the steps and formulas used to solve the problems.
Your conclusion must comprise a summary of the problems and the reason you selected a particular method to solve them. It would also be appropriate to...
...to calculate a certain term (number of months starting from January) the two previous terms must be known. These are then added together to give the desired month.
The table below shows the rabbit’s breeding numbers throughout the whole year.
The Mathematical recursive formula that represents this is:
Where: Tn= The desired month (January1, February2, March3, and so on) and where Tn>3
It can be clearly seen from the graph that the pattern/structure is exponential. This is due to the previous numbers being added in succession with the next, resulting in the ‘gap’ between each number to increase.
The trend in which the numbers follow is called a Fibonacci sequence and is often found in nature as well.
Many instances in which the Fibonacci Series is present in nature are that a lot of flowers and cone shaped structures have the number of petals as one of the Fibonacci numbers. However some plants such as the sneezewort plant (as seen left) can be seen demonstrating the Fibonacci pattern in succession. It happens on both the number of stems and number of leaves.
Another appearance of the Fibonacci Series in nature is that a lot of flowers and cone shaped structures have the number of petals as one of the Fibonacci numbers. This includes the pineapple shown to the left. The number of spirals going in each direction is a Fibonacci number. For example, there are 13...
...Questions from Questionbank
Topic 1. Sequences and Series, Exponentials and The Binomial Theorem
1. Find the sum of the arithmetic series
17 + 27 + 37 +...+ 417.
2. Find the coefficient of x5 in the expansion of (3x – 2)8.
3. An arithmetic series has five terms. The first term is 2 and the last term is 32. Find the sum of the series.
4. Find the coefficient of a3b4 in the expansion of (5a + b)7.
5. Solve the equation 43x–1 = 1.5625 × 10–2.
6. In an arithmetic sequence, the first term is 5 and the fourth term is 40. Find the second term.
7. If loga 2 = x and loga 5 = y, find in terms of x and y, expressions for
(a) log2 5;
(b) loga 20.
8. Find the sum of the infinite geometric series
9. Find the coefficient of a5b7 in the expansion of (a + b)12.
10. The Acme insurance company sells two savings plans, Plan A and Plan B.
For Plan A, an investor starts with an initial deposit of $1000 and increases this by $80 each month, so that in the second month, the deposit is $1080, the next month it is $1160 and so on.
For Plan B, the investor again starts with $1000 and each month deposits 6% more than the previous month.
(a) Write down the amount of money invested under Plan B in the second and third months.
Give your answers to parts (b) and (c) correct to the nearest dollar.
(b) Find the amount of the 12th deposit for each Plan.
(c) Find the total amount of money invested during the first 12 months
(i)...
...years, particularly that of the Fibonacci sequence and the Golden Ratio. In Debussy’s Nocturne, composed in 1892, I look into the use of the Fibonacci sequence and the Golden Ratio. Previously it has been noted that composers used the Fibonacci sequence and the Golden Ratio in terms of form, however in my analysis I look into the use of it in terms of notation as well. I will explore how the idea of Sonata form is used along with the Mathematical Model of the Fibonacci sequence. It is however important to mention that as this is one of Debussy’s earlier works, the extent that the ratio and sequence are explored are not as elaborate as some of his later works. I will explore the Harmonic analysis of the piece to create a better understanding of where and how structure is used by Debussy. Debussy was a perfectionist and would only give perfected scores to the printers, as such it is impossible to prove whether or not the use of the sequences were intended or not, however considering that some of his contemporaries in other arts were very much involved with the idea of the Golden Ratio it does seem plausible that it was intended. The fact remains, though, that the use of the sequences and ratio are still evidently there and can be analysed; as this essay will show.
The Fibonacci sequence is a system of numbers that equate to the two preceding numbers. In terms of...