Sequences and series investigation

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Sequences and series investigation By Neil In this investigation I have been asked to find out how many squareswould be needed to make up a certain pattern according to its sequence.The pattern is shown on the front page. In this investigation Ihope to find a formula which could be used to find out the numberof squares needed to build the pattern at any sequential position.Firstly I will break the problem down into simple steps to beginwith and go into more detail to explain my solutions. I will illustratefully any methods I should use and explain how I applied them tothis certain problem. I will firstly carry out this experiment ona 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 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
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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 seq. 1 2 3 4 ...

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