2D & 3D Sequences.

experiment on a 2D pattern and then extend my investigation to 3D. The Number of Squares in Each SequenceI have achieved the following information by drawing out the pattern and extending upon it.Seq. no. 1 2 3 4 5 6 7 8No. Of cubes 1 5 13 25 41 61 85 113I am going to use this next method to see if I can work out some sort of pattern:Sequence Calculations Answer1 =1 12 2(1)+3 53 2(1+3)+5 134 2(1+3+5)+7 255 2(1+3+5+7)+9 416 2(1+3+5+7+9)+11 617 2(1+3+5+7+9+11)+13 858 2(1+3+5+7+9+11+13)+15 1139 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 squares)2(3 squares) 1(5 squares)The Patterns I Have Noticied in Carrying Out the Previous MethodI have now carried out ny first investigation into the pattern and have seen a number of different patterns.Firstly I can see that the number of squares in each pattern is an odd number.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). We then take the summation (å) of these odd numbers and multiply them by two. After doing this we add on the next consecutive odd number to the doubled total.I have also noticied something through the drawings I have made of the patterns. If we look at the symetrical sides of the pattern and add up the number of squares we achieve a square number.Attempting to Obtain a Formula Through the Use of the Difference MethodI will now apply Jean Holderness' difference method to try and find a formula.Pos.in seq. 1 2 3 4 5 6 No.of squar. (c) 1 5 13 25 41 61 1st differ. (a+b) 0 4 8 12 16 20 2nd differ. (2a) 4 4 4 4 4 We can now use the equation an2 + bn + c'n' indicating the position in the sequence.If a = 2 then c = 1 and a + b = 0 If 2 is equal to b- then b = -2I will now work out the equation using the information I have obtained through using the difference method:1) 2(n -1) (n - 1) + 2n - 12) 2(n2 - 2n + 1) + 2n - 13) 2n2 - 4n + 2 + 2n - 14) 2n2 - 2n + 1Therefore my final equation is: 2n2 - 2n + 1Proving My Equation and Using it to Find the Number of Squares in Higher SequencesI will now prove my equation by applying it to a number of sequences and higher sequences I have not yet explored.Sequence 3:1. 2(32) - 6 + 12. 2(9) - 6 + 13. 18 -54. = 13The formula when applied to sequence 3 appears to be successful.Sequence 5:1. 2(52) - 10 + 12. 2(25) - 10 + 13. 50 - 10 + 14. 50 - 95. = 41 SuccessfulSequence 6:1. 2(62) - 12 + 12. 2(36) - 12 +13. 72 - 12 + 14. 72 - 115. = 61SuccessfulSequence 8:1. 2(82) - 16 + 12. 2(64) - 16 + 13. 128 - 16 + 14. 128 - 15 5. = 113 SuccessfulThe formula I found seems to be successful as I have shown on the previous page. I will now use the formula to find the number of squares in a higher sequence.So now I wil use the formula 2n2 - 2n + 1 to try and find the number of squares contained in sequence 20.Sequence 20: 2 (202) - 40 + 12(400) - 40 + 1800 - 40 + 1800 - 49= 761Instead of illustrating the pattern I am going to use the method I used at the start of this piece of coursework. The method in which Iused to look for any patterns in the sequences. I will use this to prove the number of squares given by the equation is correct.As shown below:2(1+3+5+7+9+11+13+15+17+19+21+23+25+27+29+31+33+35+37) + 39 = 761I feel this proves the equation fully.