Investigating a Sequence of Numbers.
Mathematics HL Portfolio Assignment Investigating a Sequence of Numbers [Type 1] In this Mathematics Portfolio, I am going to investigate a sequence of numbers by mathematical methods which I have learnt in the I.B. Mathematics HL course. Throughout the investigation, I will include all my workings in order to let examiners know exactly how I come up with the answers. A sequence is a set of numbers with a definite order. A series is a sum of a sequence. The sequence of numbers {an}?n =1 is: 1 x 1!, 2 x 2!, 3 x 3!, ... The two signs outside the bracket of an represent the range of the sequence. The bottom one is where the sequence begins and the one above is where it should end. Since it is stated the sequence starts from n = 1, therefore the first term, a1 = 1 x 1!, the second term, a2 = 2 x 2! and the third term, a3 = 3 x 3!...... The ! sign after the numbers is called a factorial notation. The notation basically means the product of all the numbers from 1 to the number with the notation. For example: 3! = 1 x 2 x 3 = 6 5! = 1 x 2 x 3 x 4 x 5 = 120 ? n! = 1 x 2 x 3 x 4 x ...... x n 2! x 3 = 1 x 2 x 3 = 3! = 6 ? n! x (n + 1) = (n + 1)! Going back to the investigation, to find the nth term of the sequence, the steps are shown below: a1 = 1 x 1! = 1 a2 = 2 x 2! = 2 x 1 x 2 = 4 a3 = 3 x 3! = 3 x 1 x 2 x 3 = 18 a4 = 4 x 4! = 4 x 1 x 2 x 3 x 4= 96 . . . ? an
Is there maths behind M.C. Escher’s work? If so, what elements are there?
Is there maths behind M.C. Escher's work? If so, what elements are there? In this essay, before I start anything, I must first clarify that I deeply consider mathematics as a subject that has had a great influence on the artist and his masterpieces, therefore I already alarm you that throughout my essay I will talk about Escher's work and try to persuade you that there has been a considerable integration of the subject matter with his very artworks. In order to make you understand my objective, I have gathered some of his work, then selected a few, which I found had more mathematical elements, then with a decreased amount of drawings to work with, I would be able to study all components and show you that there has been a great influence of maths on him. I believe these images without the existence of any mathematical aspect would not be able to be fully accomplished. Elements like: symmetry (reflection also included), pattern/tessellation (repetition), transformation, crystallography, "impossible shapes", proportion and the 'Fibonacci Sequence' or the 'Golden Ratio'. Are suggested to be present in M.C. Escher's artworks, these I believe have been responsible to create the effect they create on the viewer, which is wonder and marvellous of the impressive art that cannot belong to the real world. Later on I will mention and try to explain these components, so that a random
Round and Round -nà(+1) à(¸2) à
GCSE Maths Coursework: Round and Round Introduction I have been asked to investigate the equation - n-->(+1) -->(?2) --> * ? (a) (b) I will do this by first of all, changing the first number (a) to find out if that it has any relevance to the answer that comes out of the equation at the end. Then I am going to change the (b) number to find out weather that has anything to do with the outcome of the final number, I will also be looking for patterns and sequences in the answers. Investigation 1 6-->(+1)-->(?2)--> = 3.5 2.25 1.625 1.3125 1.15625 1.078125 1.0390625 1.01953125 1.009765625 1.004882813 1.002441406 1.001220703 1.000610352 1.000305176 As you can see, when the sum is entered in to a graphical calculator, the numbers that come out are as above. The numbers are decending from 3.5 to 1.000305176, that is the most that I have done down to so far. I predict that the numbers will eventually stop ay the number 1 as near the end there is three 0's and before that there were two 0's and before that there was one 0, so I estimate that there will eventually be nine 0's and the number will be finally 1.000000000. Investigation 1:carried on 1.000152588 1.000076294 1.000038147 1.000019074 1.00009537 1.000047685 1.000023843 1.000011922 1.000005961 1.000029805 1.000014903 here an extra "0" is
Maths Investigation - Pile 'em High
Maths Investigation - Pile 'em High "Jo has started in a local supermarket and her first job is to build displays of soup tins. To make them stable they are stacked using a brick bond so that each tins stands on two others. The tins are stacked against the wall. Each stack is complete with one tin in the top row." This piece of Maths coursework is about investigating sequences from a practical situation. In this investigation, tins are used to build stacks using a brick bond so that each new tin stands on two others. The tins are stacked flat against a structure and each stack is complete with one tin in the top row. An Example: First I will investigate a two row stack. In this two row stack there are: Two tins on the base (row 2) One tin on the top (row 1) Three tins altogether In a three row stack there are: Three tins on the base (row 3) Two tins in the middle (row 2) One tin on the top (row 1) Six tins altogether An early pattern that I can see is that whatever the row number (counting down from row 1 at the top) there are that many tins in the row e.g. row 3 - 3 tins. It appears to me that there is some kind of pattern forming with the total number of tins in each built up stacks. (See table) Number of Stacks The Amount of Tins 2 3 3 6 4 0 5 ? I predict that for five stacks, the amount of tins needed will be fifteen based on other stacks e.g.
The Towers of Hanoi.
The Towers of Hanoi The Towers of Hanoi is a mathematical game. It consists of 3 poles in alphabetical order and 3-7 discs numbered in size order. At the beginning of the game all the discs are on pole A in size order. The object of the game is to transfer the entire amount of discs from pole A to pole B or pole C in the minimum amount of moves possible. These are the rules of the towers of Hanoi: * A disc must never be placed on top of a smaller disc than its self. * Only one disc may move at a time. * The discs must be on a pole at all times except for one moving Discs Moves 3 7 4 5 5 31 6 63 7 27 By looking at the table it is clear there is a pattern. Every move that is made doubles the last number of moves and adds one. The algebra equation: 2x+1. (Where x is the previous amount of moves) The tower with 3 discs makes 7 moves. Therefore using the rule 2x+1 x = 7 2*7 + 1= 15 = no. moves made in a 4 disc tower. Each time a tower is built it follows the same pattern as the one before as it uses the same moves to produce a Hanoi tower until the additional disc is moved. Then the additional disc moves once and the rest of the discs repeat, building the tower but on the other pole to form the Hanoi tower from that of which the final Hanoi tower is built. The tower is built the same as the previous tower in the sequence. Although after the
Analyse the title sequences of two TV programmes, comparing and contrasting the techniques used are their effects on the audience
Analyse the title sequences of two TV programmes, comparing and contrasting the techniques used are their effects on the audience A television title sequence has to carry out a number of important roles. Firstly, the signature tune signifies which programme is about to start. Some well known signature tunes are instantly recognisable, even to young children. The music in a title sequence is like a scent. It attracts its viewers to the television in the same way bees are attracted to pollen. Secondly, the title sequence aims to be associated with the programme. The signature tune has the role of giving an insight into the programme's style and a good signature tune provides clues as to what the programme is about, for example a ghost story may have an atmospheric signature tune. In addition, the title sequence can be used to introduce the programme's characters. Furthermore, it is intended to attract and maintain viewer's attention and finally, the title sequence acts as a 'wrapper' around the programme. It is a vital part of the whole package. 'The Bill' and 'NYPD Blue' are the two television title sequences to be compared. As both programmes are crime related dramas, it will be interesting to investigate whether both title sequences create similar expectations of the proceeding programme. The title sequence of 'The Bill' opens with a close up shot of bright blue
I'm going to investigate the difference between products on a number grid first I'm going to draw a box round four numbers then I will find the product of top left, bottom right numbers, and then
Introduction: I'm going to investigate the difference between products on a number grid first I'm going to draw a box round four numbers then I will find the product of top left, bottom right numbers, and then I'm going to do the same with the top right, bottom right numbers in the box. I'll will then calculate the difference between these products. Not only I am going to investigate squares, but also rectangles, I'll use algebra to try and find a formula or a general rule that will give me this difference for any size shape. I'm going to draw a box round four numbers then I will find the product of top left, bottom right numbers, and then I'm going to do the same with the top right, bottom right numbers. 2 3 22 23 The difference between 286 and 276 is 10 because 286 - 276 = 10 13 x 22 = 286 2 x 23 = 276 49 50 59 60 The difference between 2950 and 2940 is 10 because 2950 - 2940 = 10 50 x 59 = 2950 49 x 60 = 2940 5 6 5 6 The difference between 90 and 80 is 10 because 90 - 80 = 10 6 x 15 = 90 5 x 16 = 80 I predicted that the difference for all 2x2 spares will be 10, I'll do another 2 by 2 grid to confirm that my prediction is correct. 82 83 92 93 The difference between 7636 and 7626 is 10 because 7636 - 7626 = 10 83 x 92 = 7636 82 x 93 = 7626 This shows that my prediction
Explaining the Principle of mathematical induction
Explaining the Principle of mathematical induction: Formally the principle of a proof by induction can be stated as follows: A proposition P (n) involving a positive integer n, is true for all positive integral values of n if, P (1), and P (k) ? P (k +1) is true. This can be explained using a staircase as a simple analogy. Image the proposition that a man can climb a given uniform staircase, to prove this statement we need to show two things. These are that the man can get onto the first step and that he is able to climb from one step to an other. Now relating this to the formal principle of induction, the staircase can be considered the general proposition P (n). The first step of the staircase is P (1), the second P (2), the third P (3), and so on. If we can show that the man can get onto the first step P (1) then we have ironically finished the first step of proving the proposition. The second step would be to prove that he can get from one step to an other formally put P (k) ? P (k +1). If we can show this than it follows that man can get from the first to the second step, second to the third,... n steps. Thus it can be said that P (n) is true for all positive integral values of n. 2 Definition of the derivative function f (x) The derivative function is a general expression for the gradient of a curve at any given point. It is based on the principle of limiting
My investigation is about the Phi Function .
My investigation is about the Phi Function ?. I am investigating the different ways on how to find the Phi Functions of different numbers and finding easier ways of finding the Phi Functions of large numbers. I will go through four parts for this coursework. I will start from the simplest cases of numbers and will go to more complicated. For any positive integer n, the Phi function ?(n) is defined as the number of positive integers less than n which have no factor (other than 1) in common (are co-prime) with n: So ?(12)=4, because the positive integers less than 10 which have no factors other than 1, in common with 12 are 1, 5, 7, 11 i.e. 4 of them. These four numbers are not factors of 12. Also ?(6)=2, because the positive integers less than 6 which have no factors other than 1, in common with 6 are 1, 5 i.e. 2 of them. These two numbers are not factors of two. For the first part I will find the Phi Functions of many simple numbers and I will try to find a pattern on the Phi Functions of different types of numbers e.g. Odd numbers, even numbers, prime numbers, squared numbers, triangular numbers and so on. I will start from the numbers I obtained from the coursework sheet. . ?(3)=1, 2. Two of the numbers are not the factors of 3. So ?(3)=2 2. ?(8)=1, 2, 3, 4, 5, 6, 7. Four of the numbers are not factors of 8. So ?(8)=4 3. ?(11)=1, 2, 3, 4, 5, 6, 7, 8, 9, 10. Ten of
Maths Coursework: Consecutive Numbers
MARTIN GALLAGHER Maths Coursework: Consecutive Numbers Introduction In this coursework I am trying to find a common pattern in consecutive numbers, and then solve the pattern with algebra. Problem 1: Write down three consecutive numbers. Square the middle number; multiply the first number by the third. Compare your answers, what do you notice? Problem 2: Write down two consecutive numbers. Square both of the numbers and find the difference between the two squares, what do you notice? Problem 3: Write down five consecutive numbers. Square the middle number; multiply the first number by the fifth. Compare your answers, what do you notice? Problem 4: Write down three consecutive numbers but this time with a gap between them, like this: 1, 3, 5. Square the middle number; multiply the first number by the third. Compare your answers, what do you notice? Problem 1 Chosen consecutive numbers Middle number squared First number multiplied by the last. Difference 9, 10, 11 5, 6, 7 7, 8, 9 0, 11, 12 8, 19, 20 00 36 64 21 361 99 35 63 20 360 In the above table you can see that each set of consecutive numbers created shows the "Middle number squared" is +1 more than the "First number multiplied by the last". From this pattern I will solve it by using algebra. The equation for the "Chosen consecutive numbers" is: n, n + 1, n + 2. Formula n, n + 1, n + 2 =