GCSE: Consecutive Numbers
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GCSE Maths questions
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 Level: GCSE
 Questions: 75

Chessboard coursework
4. Once you have completed this you should have 208 squares on the 8x8 grid. Further calculations 1x1=1 2x2=5 3x3=14 4x4=30 5x5=......... 1x1 1x1=1 =1 2x2 2x2=4 (21)x(21)=1x1=1 =4+1=5 3x3 3x3=9 (31) (31)=2x2=4 (21) (21)=1x1=1 =9+4+1=14 4x4 4x4=16 (41)x(41)=3x3=9 (31)x(31)=2x2=4 =16+9+4+1=30 (21)x(21)=1x1=1 Check 4x4 2 2 2 2 = (4) + (41) + (31) + (21) = 16+9+4+1 =30 Use of Algebra 2 2 2 2 (nxn) = (n) + (n1) + (n2) + (n3) Estimating 5x5 2 2 2 2 2 = (4) + (51) + (52) + (53) + (54) = 25+16+9+4+1 = 55 Table of values x y D1 D2 D3 1 1 +4 2 5 +5 +9 +2 3 14 +7
 Word count: 1077

Matrix Powers
Therefore the value of M4= d) To calculate the value for matrix 'M' when n=5, the matrix must be multiplied by an exponent of 5. Therefore the value of M5= e) To calculate the value for matrix 'M' when n=10, the matrix must be multiplied by an exponent of 10. Therefore the value of M10= f) To calculate the value for matrix 'M' when n= 20, the matrix must be multiplied by an exponent of 20. Therefore the value of M20= g)
 Word count: 1774

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
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 is correct, that all 2 by 2 will equal to 10. 3 by 3 I'm going to draw a box round nine 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.
 Word count: 1292

Fraction Differences
56 72 90 First Difference 4 6 8 10 12 14 16 18 Second Difference 2 2 2 2 2 2 2 2 As there was a constant difference of 2 I believed that the formula would include n�. I applied this to the first number in the sequence '2'. So n� = (1 x 1 = 1). To get the first number of the sequence  2 I would have to add 1. Therefore the formula could be: n2 + 1 I tried this formula for the second number in the series.
 Word count: 1564

Borders and squares
I also predict that in this project we will get the formula (2n2)  2n+1. Now I am going to draw the diagrams: 1 2 3 4 5 6 I have achieved the following information by drawing out the pattern and extending upon it. Seq. no 1 2 3 4 5 6 No. Of cubes 1 5 13 25 41 61 I am going to use this next method to see if I can work out some sort of pattern: 1 5 13 25 41 61 1st difference 4 8 12 16 20 2nd difference +4 +4 +4 +4 From the patterns I have carried out I have noticed: > That this pattern is a changing difference.
 Word count: 1702

Maths Investigation  Pile 'em High
(See table) Number of Stacks The Amount of Tins 2 3 3 6 4 10 5 ? I predict that for five stacks, the amount of tins needed will be fifteen based on other stacks e.g. for two stacks there are three and then for three stacks there are six so two tins are added. Then as you go down the table the tins adds on another i.e. +2, +3, +4. I have tested it practically using real tins and for five stacks there were fifteen tins.
 Word count: 1453

Analyse the title sequences of two TV programmes, comparing and contrasting the techniques used are their effects on the audience
The title sequence of 'The Bill' opens with a close up shot of bright blue flashing lights, which signifies an emergency. Black and white chequered tape rolls across the screen in a suspended edit and then the viewer is immediately informed that a crime drama is about to start. An atmosphere of danger and excitement is created ensuring the viewer wants to keep watching. A car is seen hurtling across the screen in an attempt to involve the audience. The camera emphasises the dramatic nature of this by magnifying the image of the car.
 Word count: 1336

Maths  Baker's Dozen
In the second question I intend to a formula to calculate the number of swaps for any number of buns. To do this I had to investigate how many swaps are needed with different amount of buns e.g. two of each bun and six of each bun. I used the same method that was used above, in question 1. Here however I will show one of each bun to five of each bun as I feel that it is a sufficient amount to show.
 Word count: 1660

Investigating a Sequence of Numbers.
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 .
 Word count: 1550

Investigate the sequence of squares in a pattern.
1 1+3+1 5 2 1+3+5+3+1 13 3 1+3+5+7+5+3+1 25 4 1+3+5+7+9+7+5+3+1 41 5 1+3+5+7+9+11+9+7+5+3+1 61 6 1+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 addition, I have noticed that the numbers that make up the total are odd. If you take the first sequence number, you can notice that the maths in the answer is 1+3+1 = 5. Secondly I can see that the number of squares in the pattern can be found out by taking the odd numbers from 1 onwards and adding them up (according to the sequence).
 Word count: 1642

Borders  a 2 Dimensional Investigation.
Total number of squares (tn) 1 1 2 5 3 13 4 25 5 41 6 61 To work out which formula to use I will now put my results in a table showing the differences between the numbers: n tn 1st difference 2nd difference 1 1 4 2 5 4 8 3 13 4 12 4 25 4 16 5 41 4 20 6 61 I can clearly see that the 2nd difference is the same, which means that my sequence is a quadratic sequence.
 Word count: 1104

Study the topic of trios and work on from that, to discover patterns and links.
2 2 3 1 1 5 1 2 4 1 3 3 1 4 2 3 1 3 2 4 1 3 3 1 2 1 4 4 2 1 4 1 2 Trios for 8: 1 2 5 1 5 2 2 5 1 2 1 5 5 1 2 5 2 1 1 3 4 1 4 3 3 4 1 3 1 4 4 3 1 4 1 3 6 1 1 1 6 1 1 1 6 2 2 4 2 4 2 4 2 4 2 3 3 3 2 3 3 3 2 Numbers
 Word count: 1946

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.
 Word count: 1403

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
 Word count: 1971

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.
 Word count: 1261

Round and Round nà(+1) à(¸2) à
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
 Word count: 1513

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 2n1 2x91=17 I will also use a sum I have not previously worked out. 65+66=131 2n1 2x661=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 +
 Word count: 1848

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 +
 Word count: 1011

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.
 Word count: 1626

Is there maths behind M.C. Escher’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'.
 Word count: 1523

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.
 Word count: 1276

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)
 Word count: 1127

A Square Investigation
16 3 9 14 196 19 3 6 15 225 22 3 5 16 256 25 3 6 17 289 28 3 9 18 324 32 4 4 19 361 36 4 1 20 400 40 4 0 21 441 44 4 1 22 484 48 4 4 23 529 52 4 9 24 576 57 5 6 25 625 62 5 5 26 676 67 5 6 27 729 72 5 9 28 784 78 6 4 29 841 84 6 1 30 900 90 6 0 31 961 96 6 1 32 1024 102 6 4 33 1089
 Word count: 1041