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

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  • Level: GCSE
  • Subject: Maths
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The Towers of Hanoi is an ancient mathematical game. The aim of this coursework is to try to identify patterns and rules associated with the game and explain them in mathematical terms.

Maths Coursework Robert Allen Queen Elizabeth Grammar School The Towers of Hanoi The Towers of Hanoi is an ancient mathematical game. The aim of this coursework is to try to identify patterns and rules associated with the game and explain them in mathematical terms. The definitions and rules are: Rules: * There are only three positions a disc can be placed. Poles A, B or C. * A disc can only go on top of a larger one. (I.e. Disc A can only go on top of Discs B and C, but Disc B cannot go on top of disc A) * The object of the game is to get all the discs to move from pole A to pole B of C in the least number of moves. * Only one disc may be moved at a time. Finding Formula A Number Of Discs Least Number Of Moves Previous term (Doubled) 2 3 2 3 7 6 4 5 4 5 31 30 6 63 62 7 27 26 8 255 254 From looking at the table it is quite clear that there is a pattern linking the number of discs and the least number of moves. It is clear that there is an element of doubling involved, as the least number of moves nearly doubles each time. When I add the extra column see above, it is clear that there is a doubling element involved. When I look again, I can see that the pattern is the previous term doubled plus 1. This can be expressed mathematically as: Un = 2(Un-1) +1 This can be shown in: . For 1 disc, it takes 1 move to move disc A from pole 1

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  • Level: GCSE
  • Subject: Maths
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Investigate the sequence of squares in a pattern.

Borders Coursework Aim: To investigate the sequence of squares in a pattern as shown below: In this investigation, I have been asked to find out how many squares would be needed to make up a certain pattern according to its sequence. In this investigation I hope to find a formula which could be used to find out the number of squares needed to build the pattern at any sequential position. Firstly I will break the problem down into simple steps to begin with and go into more detail to explain my solutions such as the nth term. I will illustrate fully any methods I should use and explain how I applied them to this certain problem. I will firstly carry out this experiment on a 2D pattern and then extend my investigation to 3D. Apparatus: Variety of sources of information A calculator A pencil A pen Paper Ruler A computer to work out equations on I have come up with the following numbers and sequences. This was done by drawing out the sequence. Seq no 2 3 4 5 6 7 No of squares 5 3 25 41 61 85 I will use these numbers to try to create a type of formula to get any no of squares in any sequence. +3+1 5 2 +3+5+3+1 3 3 +3+5+7+5+3+1 25 4 +3+5+7+9+7+5+3+1 41 5 +3+5+7+9+11+9+7+5+3+1 61 6 +3+5+7+9+11+13+11+9+7+5+3+1 85 Firstly, I have noticed that if you take the patterns, you can notice that the patterns go up by intervals of two. In

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  • Level: GCSE
  • Subject: Maths
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Find out how many squares would be needed to make up a certain pattern according to its sequence.

In this experiment I am going to require the following: A calculator A pencil A pen Variety of sources of information Paper Ruler In this investigation I have been asked to find out how many squares would be needed to make up a certain pattern according to its sequence. The pattern is shown on the front page. In this investigation I hope to find a formula which could be used to find out the number of squares needed to build the pattern at any sequential position. Firstly I will break the problem down into simple steps to begin with and go into more detail to explain my solutions. I will illustrate fully any methods I should use and explain how I applied them to this certain problem. I will firstly carry out this experiment on a 2D pattern and then extend my investigation to 3D. The Number of Squares in Each Sequence I have achieved the following information by drawing out the pattern and extending upon it. Seq. no. 1 2 3 4 5 6 7 8 No. Of cubes 1 5 13 25 41 61 85 113 I am going to use this next method to see if I can work out some sort of pattern: Sequence Calculations Answer =1 1 2 2(1)+3 5 3 2(1+3)+5 13 4 2(1+3+5)+7 25 5 2(1+3+5+7)+9 41 6 2(1+3+5+7+9)+11 61 7 2(1+3+5+7+9+11)+13 85 8 2(1+3+5+7+9+11+13)+15 113 9 2(1+3+5+7+9+11+13+15) +17 145 What I am doing above is shown with the aid of a diagram below; If we take sequence 3: 2(1+3)+5=13 2(1

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  • Level: GCSE
  • Subject: Maths
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Continued Fractions

Continued Fractions This 'infinite fraction' can be considered as a sequence of terms, tn: A general formula for tn+1 in terms of tn can now be determined. It can be seen that tn+1 is 1added to 1 divided by the previous term. i.e. Decimal equivalents of each term can be computed. Here are the values for the first ten terms (correct to 5d.p as we can then see how the numbers differ): t1=1 t2= 1.50000 t3= 1.66667 t4= 1.60000 t5= 1.62500 t6= 1.61538 t7= 1.61905 t8= 1.61765 t9= 1.61818 t10= 1.61798 From looking at the graph we can see that for the first few terms the values fluctuate, but eventually the values fluctuate less and become very close together. I.e. the values become closer together as the value of n increases. We can then conclude that as n increases tn˜tn+1. From this, we can now deduce a formula for tn+1 in terms of tn: We can also conclude that if tn˜tn+1, then tn-tn+1 will equal to zero. Any term (the nth term) can be determined using the above general formula (formula 1), but it requires having to calculate all the values up to that term. For example, the 200th term can be found but we would have to find all the values up to the 199th term. This is time-consuming and instead of having to do this, we can find a new general formula: (we can assume that tn=tn+1 when n is a large value) (this makes it easier to understand the origins of the

  • Word count: 825
  • Level: GCSE
  • Subject: Maths
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Beyond Pythagoras

a) The numbers 5,12,13 satisfy the condition a2 + b2 = c2 52 + 122 = 132 52 = 5 x 5 = 25 122 = 12 x 12 = 144 132 = 13 x 13 = 169 and so 52+ 122 = 25 + 144 = 169 = 132 b) The numbers 7, 24, 25 a2 + b2 = c2 72 + 242 = 25 because 72 = 7 x7 = 49 242 = 24 x 24 = 576 252 = 25 x 25 = 625 and so 72 + 242 = 49 + 576 = 625 = 25 2 The numbers above satisfy similar condition that is pythagors theorem. I have tested the numbers by putting them into the formula a2 + b2 = c2. This is means the numbers are a Pythagorean triple because they satisfy the condition a2 + b 2 = c2 2) Length of shortest side Length of middle side Length of longest side Perimeter Area 3 4 5 2 6 5 2 3 30 30 7 24 25 56 84 9 40 41 90 80 a) I found the perimeter for the sequence 5, 12, 13 and 7, 24, 2, by simply finding the sum (add) all the lengths of the three sides together to get the perimeter. E.g. Perimeter = shortest +middle + longest length 5 + 12 + 13 = 30 units 7 + 24 + 25 = 56 units I found the area for the sequences 5, 12, 13 and 7, 24, 25, by simply finding the product (multiply) the shortest and middle and halving the answer. E.g. Area = 1/2 x shortest x middle length 1/2 x 5 x 12 = 30 square units 1/2 x 7x 24 = 84 square units I used this method for the area because it is a right-angled triangle. The dotted lines above show that it is actually

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  • Level: GCSE
  • Subject: Maths
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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

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  • Level: GCSE
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Number Grid Investigation.

Number Grid Investigation I was first given A 10x10 grid, counting from 1-100. Inside the grid was A 2x2 box surrounding the numbers, 12, 13, 22 and 23; 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 00 I was asked to; * Find the product of the top left number and bottom right number in the box. * Do the same with the top right and bottom left numbers n the box. * Calculate the difference between these numbers. I did this, 2 x 23 = 276 3 x 22 = 286 286 - 276 = 10 The difference between them was 10 I decided to try it with A 3x3 box surrounding the numbers; 2 3 4 22 23 24 2 x 34 = 408 4 x 32 = 448 448 - 408 = 40 The difference between them was 40 I also tried this with A 4x4 box, and A 5x5 box: (4x4) 12 x 45 = 540 15 x 42 = 630 630 - 540 = 90 (5x5) 12 x 56 = 672 16 x 52 = 832 832 - 672 = 160 I decided to try and find A pattern between these numbers, I thought that since the boxes grows in an even number, so should the totals. Put into order they looked like this: Side of box length 2 3 4 5 Difference in

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  • Level: GCSE
  • Subject: Maths
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Towers of Hanoi.

Maths GCSE Coursework Task Towers of Hanoi Introduction We have been asked during this piece of coursework to investigate the Towers of Hanoi. The Towers of Hanoi is a simple game whereby you must move of a pile of 3, 4, 5 or any other number of discs (1, 2, 3, etc) of decreasing radii from 1 of 3 poles to another pole (A, B, C). You are only able to move one disc at a time and cannot place a larger disc on top of a smaller disc. You must also complete this task in the smallest amount of moves possible. Our ultimate task was to complete the game with 4 discs and then 5 discs using the smallest amount of moves, then to find a formula to find the smallest amount of moves for any number of discs. Simple cases: 4 discs After having tried to solve the puzzle with 4 discs I found that the smallest amount of moves possible was 15. (See fig. 1) 5 discs To try and make things slightly easier for myself I decided to use the first 15 moves I had used for 4 discs and then proceed from there. This method was effective and led me to find that the smallest number of moves was 31. (See fig. 2) Results and Formulas Number of discs Number of moves 2 3 3 7 4 5 5 31 When placing all the results into a table I noticed that if you take a certain number of moves for example 3 and then double it you end up with 6. You only then need to add another 1 to make 7, which is the

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  • Level: GCSE
  • Subject: Maths
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Towers of Hanoi

Towers of Hanoi Introduction: Towers of Hanoi is a puzzle/game in which you have to move a certain number of discs from pole a to pole c in the minimum amount of moves possible. There are a few certain rules you have to follow though: . You CANNOT move 2 discs at a time 2. You CANNOT place a smaller disc over a bigger disc. Aim: The main of the investigation is to enquire into the relationship between the number of discs and the minimum number of moves to complete. We also have to investigate the symmetry of where the disc's go and move to the poles. Also number of moves made by the individual discs to be moved at each stage of the process. We also have to find a formula linking the minimum no of moves to the number of discs. How I went about finding this data: When we were told to investigate Towers of Hanoi we had to figure how many moves it would take to get to pole C in the least amount of moves for 4 discs. So I got a plain piece of paper and drew the three poles on it. Then I got 4 different coloured pieces of paper from the teacher and sat down and tried figuring it out. I also played the game on the schools computers. It was more effective playing it on the computer because you can see some animation rather than seeing bits of paper, also it doesn't seem time consuming when on the computers. The game is also on the Internet on a variety of sites but I

  • Word count: 1626
  • Level: GCSE
  • Subject: Maths
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