My aim is to investigate Borders Essay Example

My aim is to investigate Borders Essay My aim is to investigate Borders. I will be drawing borders to different squares and finding a formula for each one and finally I will find a Universal Formula. Introduction This is what the task tells me: Here are 2 squares with squares added on each side to make a border, which surrounds the starting squares. We will write a custom essay sample on My aim is to investigate Borders specifically for you for only $16.38 $13.9/page Order now We will write a custom essay sample on My aim is to investigate Borders specifically for you FOR ONLY $16.38 $13.9/page Hire Writer We will write a custom essay sample on My aim is to investigate Borders specifically for you FOR ONLY $16.38 $13.9/page Hire Writer You can then add another border as shown: Investigate Borders. Method First I will find out how many squares needed for the border to a 11 square, then 21 and so on up to 51. Then I will find a formula for the border to each square and also test the formula out to prove that it works. I will predict how many squares needed for the 6th border and find out if my prediction was correct. Next, I will find out how many squares needed for the border of a 12 square, then 22, and so on up to 52, then 13, 23, and so on up to 53. Again I will find formulas to them and prove that all the formulas work. Finally, I will put all the formulas together in three groups and find one overall formula for each of the groups. Then, I will get those three formulas and get one Universal formula in the end. Diagram of Borders of square: 11 Table of results for Borders of square: 11 Formula You can always find the nth term using the Formula: a is simply the value of THE FIRST TERM in the sequence. d is simply the value of THE COMMON DIFFERENCE between the terms. To get the nth term you just need to find the values of a and d from the sequence and stick them in the formula. Formula to find the number of squares needed for each border (for square 11): Common difference = 4 First term = 4 Formula = Simplification = Experiment I will try to find the number of squares needed for border number 6 using the formula, I found out, above: nth term = 4 x 6 = 24 Common Difference nth Term Results My prediction was 24 which is the correct answer, for the number of squares needed for border number 6, and which also proves that the formula is in working order. Diagram of Borders of square: 21 Table of results for Borders of square: 21 Formula to find the number of squares needed for each border (for square 21): Common difference = 4 First term = 6 Formula = Simplification = Experiment I will try to find the number of squares needed for border number 6 using the formula, I found out, above: nth term = 4 x 6 + 2 = 26 Common Difference nth Term Results My prediction was 26 which is the correct answer, for the number of squares needed for border number 6, and which also proves that the formula is in working order. Diagram of Borders of square: 31 Table of results for Borders of square: 31 Formula to find the number of squares needed for each border (for square 31): Common difference = 4 First term = 8 Formula = Simplification = Experiment I will try to find the number of squares needed for border number 6 using the formula, I found out, above: nth term = 4 x 6 + 4 = 28 Common Difference nth Term Results My prediction was 28 which is the correct answer, for the number of squares needed for border number 6, and which also proves that the formula is in working order. Diagram of Borders of square: 41 Table of results for Borders of square: 41 Formula to find the number of squares needed for each border (for square 41): Common difference = 4 First term = 10 Formula = Simplification = Experiment I will try to find the number of squares needed for border number 6 using the formula, I found out, above: nth term = 4 x 6 + 6 = 30 Common Difference nth Term Results My prediction was 30 which is the correct answer, for the number of squares needed for border number 6, and which also proves that the formula is in working order. Diagram of Borders of square: 51 Table of results for Borders of square: 51 Formula to find the number of squares needed for each border (for square 51): Common difference = 4 First term = 12 Formula = Simplification = Experiment I will try to find the number of squares needed for border number 6 using the formula, I found out, above: nth term = 4 x 6 + 8 = 32 Common Difference nth Term Results My prediction was 32 which is the correct answer, for the number of squares needed for border number 6, and which also proves that the formula is in working order. Formula to find the number of squares needed for each border (for square 12): Formula = Simplification = I have already found out the Formula for 12 so theres no need for the Diagrams ; Tables, and I have already proved that the Formula works. Diagram of Borders of square: 22 Table of results for Borders of square: 22 Formula to find the number of squares needed for each border (for square 22): Common difference = 4 First term = 8 Formula = Simplification = Experiment I will try to find the number of squares needed for border number 6 using the formula, I found out, above: nth term = 4 x 6 + 4 = 28 Common Difference nth Term Results My prediction was 28 which is the correct answer, for the number of squares needed for border number 6, and which also proves that the formula is in working order. Diagram of Borders of square: 32 Table of results for Borders of square: 32 Formula to find the number of squares needed for each border (for square 32): Common difference = 4 First term = 10 Formula = Simplification = Experiment I will try to find the number of squares needed for border number 6 using the formula, I found out, above: nth term = 4 x 6 + 6 = 30 Common Difference nth Term Results My prediction was 30 which is the correct answer, for the number of squares needed for border number 6, and which also proves that the formula is in working order. Diagram of Borders of square: 42 Table of results for Borders of square: 42 Formula to find the number of squares needed for each border (for square 42): Common difference = 4 First term = 12 Formula = Simplification = Experiment I will try to find the number of squares needed for border number 6 using the formula, I found out, above: nth term = 4 x 6 + 8 = 32 Common Difference nth Term Results My prediction was 32 which is the correct answer, for the number of squares needed for border number 6, and which also proves that the formula is in working order. Diagram of Borders of square: 52 Table of results for Borders of square: 52 Formula to find the number of squares needed for each border (for square 52): Common difference = 4 First term = 14 Formula = Simplification = Experiment I will try to find the number of squares needed for border number 6 using the formula, I found out, above: nth term = 4 x 6 + 10 = 34 Common Difference nth Term Results My prediction was 34 which is the correct answer, for the number of squares needed for border number 6, and which also proves that the formula is in working order. Formula to find the number of squares needed for each border (for square 13): Formula = Simplification = As before, I have already done the working out for this Formula and it has been proved. Formula to find the number of squares needed for each border (for square 23): Formula = Simplification = Again the working outs been done. Diagram of Borders of square: 33 Table of results for Borders of square: 33 Formula to find the number of squares needed for each border (for square 33): Common difference = 4 First term = 12 Formula = Simplification = Experiment I will try to find the number of squares needed for border number 6 using the formula, I found out, above: nth term = 4 x 6 + 8 = 32 Common Difference nth Term Results My prediction was 32 which is the correct answer, for the number of squares needed for border number 6, and which also proves that the formula is in working order. Diagram of Borders of square: 43 Table of results for Borders of square: 43 Formula to find the number of squares needed for each border (for square 43): Common difference = 4 First term = 14 Formula = Simplification = Experiment I will try to find the number of squares needed for border number 6 using the formula, I found out, above: nth term = 4 x 6 + 10 = 34 Common Difference nth Term Results My prediction was 34 which is the correct answer, for the number of squares needed for border number 6, and which also proves that the formula is in working order. Diagram of Borders of square: 53 Table of results for Borders of square: 53 Formula to find the number of squares needed for each border (for square 53): Common difference = 4 First term = 16 Formula = Simplification = Experiment I will try to find the number of squares needed for border number 6 using the formula, I found out, above: nth term = 4 x 6 + 12 = 36 Common Difference nth Term Results My prediction was 36 which is the correct answer, for the number of squares needed for border number 6, and which also proves that the formula is in working order. The overall formula to the borders of nx1: 11 = 4n 21 = 4n + 2 31 = 4n + 4 41 = 4n + 6 51 = 4n + 8 Formula for nx1th term: 4 y + ( 2 n 2 ) e.g. for border number 6: First replace n with 6. 4 y + ( 2 x 6 2 ) = 4 y + 10 Now you can replace the y with an n to have the formula, to find the number of squares, for border number 6. 4 n + 10 The overall formula to the borders of nx2: 12 = 4n + 2 22 = 4n + 4 32 = 4n + 6 42 = 4n + 8 52 = 4n + 10 Formula for nx2th term: 4 y + ( 2 n ) e.g. for border number 6: First replace n with 6. 4 y + ( 2 x 6 ) = 4 y + 12 Now you can replace the y with an n to have the formula, to find the number of squares, for border number 6. 4 n + 12 The overall formula to the borders of nx3: 13 = 4n + 4 23 = 4n + 6 33 = 4n + 8 43 = 4n + 10 53 = 4n + 12 Formula for nx3th term: 4 y + ( 2 n + 2 ) e.g. for border number 6: First replace n with 6. 4 y + ( 2 x 6 + 2) = 4 y + 14 Now you can replace the y with an n to have the formula, to find the number of squares, for border number 6. 4 n + 14 The Universal formula: The formula for Length x Width = 4 n + 2 L + 2 W 4 = B (Border) e.g. for the 6th border of a 53 rectangle: First replace n with the border number = 6, L with the Length = 5, and W with the Width = 3. Then add brackets where necessary. ( 4 x 6 ) + ( 2 x 5 ) + ( 2 x 3) 4 = B Then multiply out the brackets: 24 + 10 + 6 4 = 36 36 is the correct answer. Formulas for nx1 11 = 4n 21 = 4n + 2 31 = 4n + 4 41 = 4n + 6 51 = 4n + 8 Formulas for nx2 11 = 4n + 2 21 = 4n + 4 31 = 4n + 6 41 = 4n + 8 51 = 4n + 10 Formulas for nx3 11 = 4n + 4 21 = 4n + 6 31 = 4n + 8 41 = 4n + 10 51 = 4n + 12 Conclusion In the time available to me, I believe I have researched Borders to the full extent of my ability. I found formulas to squares nx1. I then extended this to squares nx2 and nx3, and I then was able to construct my Universal Formula, which will tell you the number of squares in any border of square nxn, which could be anything from 22 to 1015. I also found that many of my predictions I made along the way turned out to be correct. I would say that this investigation has been a success. I began this investigation with the aim to find formulas to nx1, nx2 nx3 and then a Universal Formula and they were achieved.