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# GCSE: Consecutive Numbers

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Meet our team of inspirational teachers Get help from 80+ teachers and hundreds of thousands of student written documents 1. ## This coursework will be to investigate to see how many squares would be needed to make any cross-shape build up in this way.

4 4 No of squares: 1 5 13 25 41 61 1st Diff: 4 8 12 16 20 2nd Diff: 4 4 4 4 To find the 1st and 2nd difference, I used a table and then the diagram above to check if I was right about my suggested answers. First of all, for the 1st difference, I took the number of squares for shape number 2 and subtracted that with the number of squares I got for the first shape, which then gave me the 1st difference.

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2. ## The Towers of Hanoi.

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 tower is built the new disc is moved to pole B or C. This move is where the plus one comes from the equation as it is a separate move that is not a part of the unbuilding or rebuilding process of the previous tower.

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3. ## In this investigation I am trying to find a rule for the difference between any consecutive numbers in a sequence. I am going to use a series of algebraic expressions to try and come up with a successful rule that works for every consecutive number I try.

As you can see above, for 5 Consecutive numbers the difference goes up in 5's every time, 6 consecutive numbers the difference goes up in 6's every time, and so on... Table Of Results 2 Consecutive Numbers Sequence of Numbers Difference 1,2 3 2,3 5 3,4 7 4,5 9 3 Consecutive Numbers Sequence of Numbers Difference 1,2,3 6 2,3,4 9 3,4,5 12 4,5,6 15 4 Consecutive Numbers Sequence of Numbers Difference 1,2,3,4 10 2,3,4,5 14 3,4,5,6 18 4,5,6,7 22 5 Consecutive Numbers Sequence of Numbers Difference 1,2,3,4,5 15 2,3,4,5,6 20 3,4,5,6,7 25 4,5,6,7,8 30 6 Consecutive Numbers Sequence of Numbers

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4. ## In this investigation I will explore the relationship between a series of straight, non-parallel, infinite lines on a plane surface and analyze the number of lines, maximum number of crossover points and open and closed regions.

Diagram 2B: (open regions are depicted with numbers, crossover point is high-lighted with a red circle) Diagram 2 B represents the most regions and crossover points possible for two straight lines. Therefore I can deduce that in order to create the maximum number of crossover points and the maximum number of open and closed regions all the lines in the diagram must cross over every other line. In this diagram there are: Number of Lines (n) Cross-Over Points Open Regions Closed Regions Total Regions 1 0 2 0 2 2 1 4 0 4 Diagram 3A: Diagram 3B In Diagram

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5. ## My investigation is about the Phi Function .

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. 1. ?(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.

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6. ## Round and Round -n&agrave;(+1) &agrave;(&cedil;2) &agrave;

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 added 1.000007452 1.000003726 1.000001863 1.000000932 1.000000466 1.000000233 1.000000177 1.000000089 here one extra "0" is added. 1.000000045 1.000000023 1.000000012 1.000000006 here one more "0" is added 1.000000003 1.000000002 1.000000001 1.000000001 1.000000001 At this point, it is the furthest that the number is been to the number one, I was almost right in my prediction. I will now try a change in the sequence, I am going do the sequence as followed; 6-->(+2)-->(?2)--> = 4 3 2.5 2.25 2.125 2.0625 2.03125 2.016625 2.0078125

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7. ## To investigate consecutive sums. Try to find a pattern, devise a formulae and establish which numbers cannot be made using consecutive sums.

(All ready a pattern of +2 is clearly visible) I am now going to prove this theory. 8+9=17 2n-1 2x9-1=17 I will also use a sum I have not previously worked out. 65+66=131 2n-1 2x66-1=131 0 + 1 = 1 1 + 2 = 3 2 + 3 = 5 3 + 4 = 7 4 + 5 = 9 5 + 6 = 11 6 + 7 = 13 7 + 8 = 15 8 + 9 = 17 9 + 10 = 19 10 + 11 = 21 11 + 12 = 23 12 + 13 = 25 13 + 14 = 27 14 + 15 = 29 15 + 16 = 31 16 +

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8. ## How does the use of numbers, statistics, graphs, and other quantitative instruments affect perceptions of the validity of knowledge claims in the human sciences?

These studies on the human race are divided into many categories mainly called human sciences. These studies are called sciences because the use of numbers in collecting data and analysing them are essential to prove certain events. In todays society numbers became really important because it is an element that is essential in our life, starting from the money that we use to buy our daily needs to knowing how to count. Whenever humans encounter numbers in magazines they tend to believein them. Any quantitative instrument is believed in todays society because it is known that most of the quantitative instruments that are available are very precise.

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9. So N=N(s,t), or s=s(N) and t=t(N), where . Here I list the s and t parameters for the first five triangular squares: 1. s=1, t=1. t/s=1. 36. s=6, t=8. t/s=1.333... 1225. s=35, t=49. t/s=1.4. 41616. s=204, t=288. t/s=1.411764705... 1413721. s=1189, t=1681. t/s=1.413793103... I also listed the ratio t(N)/s(N) for each of these N. Notice how the t/s ratio approaches as N gets bigger. Also notice that t is either a perfect square or a perfect square minus one, viz.

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10. ## Transforming numbers

To do a thorough investigation, I decided to use excel spreadsheet with a > b, by 2,3,4,and so on. It always gave the same result, transformation converging towards V2. This led me to my next step to investigate what happens when a is less than b. Table3 a=5, b=7 a less than b by 2 (a<b) Sequence Sequence of of Numerator Denominator Result 5 7 0.7142857 19 12 1.5833333 43 31 1.3870968 105 74 1.4189189 253 432 1.4143519 It is seen from table 3 that, even when a is less than b, it still converges to V2.

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11. ## I am investigating how many regions can be created when n circles overlap. After I have looked at circles I will look at other shape and try to find if they have a general formula.

Term 1 2 3 4 5 Sequence 1 3 7 13 21 n* 1 4 9 16 25 New 0 -1 -2 -3 -4 Sequence 1st difference -1 -1 -1 -1 -1 And B Is the difference of the new sequence so b= -1 our formula now is Un = n*- n + c Now I have to find c we will have to put the formula into action: If n = 1 U1 = 1* - 1 + c and because u = 1 c must be +1 so the formula must be Un = n* - n +

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12. ## Towers of Hanoi

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 went on www.mazeworks.com/hanoi/index.htm which I found was the best.

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13. ## In this course work I have been asked to find out how many squares will be needed to make up a certain pattern according to its sequence.

5 8 4 3 5 8 13 12 4 4 13 12 25 16 4 5 25 16 41 20 4 6 41 20 61 24 4 7 61 24 85 28 4 8 85 28 113 32 4 9 113 32 14 5 36 4 10 145 36 181 40 4 11 181 40 221 44 4 12 221 44 265 4 Table showing my results. I have achieved these results in the table I have shown above. I got these results from the drawings I have drawn.

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14. ## Investigation to Find the number of diagonal of any 2 Dimensional or / and 3 Dimensional - A diagonal is a line drawn from one vertex (corner) of the shape to another

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.

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15. ## Beyond Pythagoras

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.

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16. ## Is there maths behind M.C. Escher&#146;s work? If so, what elements are there?

First of all, I will talk about Topology, a characteristic of maths that when used on regular images is responsible to create the 'impossible shapes', these are named this way because they are only possible to be created as pictures or as images but not as 3D shapes due to their structure: they sometimes have no real linking lines. Topology is mainly the study of a space that stays invariant even under continuous variations i.e. deformations on something, which do not create 'holes' or 'tears'.

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17. ## Explaining the Principle of mathematical induction

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 a cord on a given curve spanned between two points. As the cord gets smaller, the gradient of the cord gets closer and closer to the gradient of the tangent at the given point.

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18. ## Nth Term Investigation

The nth for this one is (n-1)2. Here are my predictions for other squares with different lengths. n x n + 10 x 10 4 36 81 25 x 25 4 96 576 50 x 50 4 196 2401 100 x 100 4 396 9801 Rectangles No.1 I will now move on to rectangles, as there are so many ways of doing rectangles I will start with one side going up by one each time (x) and the other side always being the same (t) which will be 2.

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19. ## Nth Term - Finding and verifying a formula for the nth term of a sequence.

So the formula will be 3�n + something (You can complete this something later on) This is because, if the step length is the same for all the terms in the sequence, the formula will be of the format: step � n + something For the sequence above, the rule 3�n + something would give the values 3�1 + something = 3 + something 3�2 + something = 6 + something 3�3 + something = 9 + something 3�4 + something = 12 + something 3�5 + something = 15 + something You then compare these values with the ones in the actual sequence - it should be obvious that the value of the something is +2 So the formula for the nth term is 3n + 2 That's Easy Enough!

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20. ## Consecutive Numbers Investigation

I will now hope to show that it works with algebra. X, X+1, X+2 1st *3rd = X*(x+2) = X2=2X 2nd squared = (X+1)2 = (X+1)(X+1) (X +1)(X+1) = X2 + 1 + 1X + 1X = X2 + 2X + 1 The only difference is +1. It shows that the difference will always be 1. I am now going to see what happens if I make the gap 2. Gap 2 3, 5, 7 3*7 = 21 52 = 5x5 = 25 Difference 4 5, 7, 9 5*9 = 45 72 = 7*7 = 49 Difference 4 17, 19, 21 17*21 = 357 192 = 19*19 = 361 Difference 4 It would appear that it would work every time.

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21. ## Maths Statistics Dice Investigation

6 No' time thrown 9 8 8 9 9 7 From the table above it shows that if the game was played for real it would make a small profit of �5 for every �50 that came in. There is still a 1/6 chance of winning but the prize money has been reduces so that for every 6 pounds that I took I would be paying out �5 so I would statistically be making a profit of �1 for every 6 throws that took place From the previous results that I received I think that I should increase the amount of events that can take place.

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22. ## Maths Coursework: Consecutive Numbers

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 = (n + 1)� - n(n + 2) = (n + 1)(n + 1) - (n� + 2n) = n� + n + n + 1 - (n� + 2n) = n� + 2n + 1 - (n� + 2n) = 1 Test the formula I will test my formula to prove it is correct, by replacing "n" with one of the first consecutive numbers: When "n" = 9 = (9 + 1)� - 9(9 + 2)

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23. ## Maths Coursework &#150; N Lines

10 crossovers 14 16 6 lines 1 2 3 4 5 6 7 8 9 10 13 14 11 12 16 17 18 19 20 regions 15 crossovers 12 open regions 10 closed regions 20 21 22 This is what I predicted 7 lines 1 2 3 4 5 6 11 13 14 10 15 7 8 9 17 18 19 20 29 21 22 21 Crossovers 29 regions 16 25 14 open regions 15 closed regions 23

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24. ## Number Sums Investigations

84 7+8+9+10+11+12+13+14, 9+10+11+12+13+14+15, 27+28+29 85 42+43, 4+5+6+7+8+9+10+11+12+13, 15+16+17+18+19 86 20+21+22+23 87 43+44, 12+13+14+15+16+17, 28+29+30 88 3+4+5+6+7+8+9+10+11+12+13 89 44+45 90 16+17+18+19+20, 2+3+4+5+6+7+8+9+10+11+12+13, 6+7+8+9+10+11+12+13+14, 29+30+31, 21+22+23+24 91 45+46, 1+2+3+4+5+6+7+8+9+10+11+12+13, 10+11+12+13+14+15+16, 21+22+23+24 92 8+9+10+12+13+14+15 93 46+47 94 22+23+24+25 95 47+48, 5+6+7+8+9+10+11+12+13+14, 17+18+19+20+21 96 31+32+33 97 48+49 98 23+24+25+26, 11+12+13+14+15+16+17 99 49+50, 4+5+6+7+8+9+10+11+12+13+14, 7+8+9+10+11+12+13+14+15, 14+15+16+17+18+19, 32+33+34 100 18+19+20+21+22, 9+10+11+12+13+14+15+16 Table of two number sums Number Sum Number Sum 1 0+1 51 25+26 3 1+2 53 26+27 5 2+3 55 27+28 7 3+4 57 28+29 9 4+5 59 29+30 11 5+6 61 30+31 13 6+7 63 31+32 15 7+8 65 32+33 17 8+9 67 33+34

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25. ## To see if three horizontal rectangle numbers e.g. &#147;12,13,14&#148; &#150; have the same result when you multiply the middle number by 2 and add the 1st and last number together.

when we write as it's shortest form i.e.: ( + ( = (2 ( ( 2= (2 Formula: Middle number multiplied by 2 = (First number) + (Last number) Case.2: To see if "Case.1" applies for horizontal and diagonal rectangles. Testing: Conclusion: The original formula (Case.1)

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