Find out how many squares would be needed to make up a certain pattern according to its sequence.

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 145What I am doing above is shown with the aid of a diagram below;If we take sequence 3:2(1+3)+5=132(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 andhave 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 canbe found out by taking the odd numbers from 1 onwards and addingthem up (according to the sequence). We then take the summation(Å) of these odd numbers and multiply them by two. After doing thiswe add on the next consecutive odd number to the doubled total.I have also noticied something through the drawings I have madeof the patterns. If we look at the symetrical sides of the patternand 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 6No.of squar. (c) 1 5 13 25 41 611st differ. (a+b) 0 4 8 12 16 202nd differ. (2a) 4 4 4 4 4We can now use the equationan2 + bn + c'n' indicating the position in the sequence.If a = 2 then c = 1 and a + b = 0 If2 is equal to b- then b = -2I will now work out the equation using the information I have obtainedthrough 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 Numberof Squares in Higher SequencesI will now prove my equation by applying it to a number of sequencesand 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 besuccessful.Sequence 5:1. 2(52) - 10 + 12. 2(25) - 10 + 13. 50 - 10 + 14. 50 - 95. = 41SuccessfulSequence 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. = 113SuccessfulThe formula I found seems to be successful as I have shown on theprevious 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 findthe 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 methodI used at the start of this piece of coursework. The method in whichIused to look for any patterns in the sequences. I will use thisto 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.