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 and geometric sequence because the formula made the answer easier to find the first term and the common difference. All you needed to do was find the formula and plug the numbers from the word problem to get the answer. I could apply this knowledge to real world situations because you can use this when you are checking to see how much money will be in your savings account and how much money...
...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...
...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...
...Higher Arithmetic
Higher arithmetic, also known as the theory of numbers, is known for its basics of the natural numbers, simple numbers. The numbers, 1, 2, and 3 are numbers that are known as natural numbers. H. Davenport of Cambridge University once said “…in all the records of ancient civilizations there is evidence of some preoccupation with arithmetic over and above the needs of everyday life” (Introduction). The theory of numbers being a science, is simply just a creation invented for the present times. We, as humans, learn regular arithmetic as children, with games such as marbles and other fun counting games. Eventually, as we get to elementary school, we learn the use of addition, subtraction, division, and multiplication, the basics essentially. Math tends become more complex as we move on to middle school and high school. Middle school and high school is where we eventually start using higher arithmetic to understand the current math we are being taught. Although not everyone has an actual rulebook of the higher arithmetic, these laws stand universal. A question can be asked, how important is higher arithmetic? Higher arithmetic is used daily not just by mathematicians, but people of everyday quality. Higher arithmetic is essential, however, it should not be put over human needs and the qualities of everyday life, higher...
...
This work MAT 126 Week 1 Assignment  Geometric and ArithmeticSequence 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 arithmeticsequence 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...
...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...
...Yr12 Test Nov 2009. 1.
Name:________________________
Let Sn be the sum of the first n terms of an arithmeticsequence, whose first three terms are u1, u2 and u3. It is known that S1 = 7, and S2 = 18. (a) (b) (c) Write down u1. Calculate the common difference of the sequence. Calculate u4.
(Total 6 marks)
1
2.
A theatre has 20 rows of seats. There are 15 seats in the first row, 17 seats in the second row, and each successive row of seats has two more seats in it than the previous row. (a) (b) Calculate the number of seats in the 20th row. Calculate the total number of seats.
(Total 6 marks)
2
3.
Gwendolyn added the multiples of 3, from 3 to 3750 and found that 3 + 6 + 9 + … + 3750 = s. Calculate s.
(Total 6 marks)
3
*4.
Find the term containing x10 in the expansion of (5 + 2x2)7.
(Total 6 marks)
4
*5.
Use the binomial theorem to complete this expansion. (3x +2y)4 = 81x4 + 216x3 y +...
(Total 4 marks)
5
6.
(a) (b)
Given that log3x – log3(x – 5) = log3A, express A in terms of x. Hence or otherwise, solve the equation log3x – log3(x – 5) = 1.
(Total 6 marks)
6
7.
Find the exact solution of the equation 92x = 27(1–x).
(Total 6 marks)
7
*8.
If loga 2 = x and loga 5 = y, find in terms of x and y, expressions for (a) (b) log2 5; loga 20.
(Total 4 marks)
8
9.
Solve the equation log9 81 + log9
+ log9 3 = log9 x.
(Total 4 marks)
9...
...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...
...Job sequence modeling using Genetic Algorithms
Dr.S.N.Sivanandam
Professor & Head
M.Kannan
Senior Lecturer
Department of Computer Science & Engineering,
P.S.G.College of Technology,
Coimbatore641 004
Abstract
This paper presents a Genetic algorithm (GA) based procedure for finding an optimum job sequence for N jobs / M machines problem based on minimum elapsed time. The search space is so large that the Genetic algorithms outperform the conventional procedures in solving optimization problems. In this paper we propose a Bell shaped sequence for N jobs / M machines problem. The optimum sequence resemble a Bell shaped sequence (Normal or Gaussian distribution like curve). In the sense that the maximum total processing time of a job M machines lie in the middle of the sequence, next maximum lie in the right side (or left side) and the next maximum lie in the left side (or right) and so on and an example is illustrated. By including the Bell shaped sequence in the proposed GA procedure, the convergence of optimum sequence is faster since the final optimum sequence almost always resemble a Bell shaped sequence. Various test cases are discussed in Appendix.
Keywords: Job sequencing, Genetic algorithms, Optimization, NPhard, Two Job point crossover, Mutation, Mirror image, Bell shaped sequence, Normal...