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: 3x1

Table of results for Borders of square: 3x1

Formula to find the number of squares needed for each border (for square 3x1):

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: 4x1

Table of results for Borders of square: 4x1

Formula to find the number of squares needed for each border (for square 4x1):

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: 5x1

Table of results for Borders of square: 5x1

Formula to find the number of squares needed for each border (for square 5x1):

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 1x2):

Formula = Simplification =

I have already found out the Formula for 1x2 so there’s no need for the Diagrams & Tables, and I have already proved that the Formula works.

Diagram of Borders of square: 2x2

Table of results for Borders of square: 2x2

Formula to find the number of squares needed for each border (for square 2x2):

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: 3x2

Table of results for Borders of square: 3x2

Formula to find the number of squares needed for each border (for square 3x2):

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: 4x2

Table of results for Borders of square: 4x2

Formula to find the number of squares needed for each border (for square 4x2):

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: 5x2

Table of results for Borders of square: 5x2

Formula to find the number of squares needed for each border (for square 5x2):

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 1x3):

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 2x3):

Formula = Simplification =

Again the working out’s been done.

Diagram of Borders of square: 3x3

Table of results for Borders of square: 3x3

Formula to find the number of squares needed for each border (for square 3x3):

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: 4x3

Table of results for Borders of square: 4x3

Formula to find the number of squares needed for each border (for square 4x3):

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: 5x3

Table of results for Borders of square: 5x3

Formula to find the number of squares needed for each border (for square 5x3):

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:

1x1 = 4n

2x1 = 4n + 2

3x1 = 4n + 4

4x1 = 4n + 6

5x1 = 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:

1x2 = 4n + 2

2x2 = 4n + 4

3x2 = 4n + 6

4x2 = 4n + 8

5x2 = 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:

1x3 = 4n + 4

2x3 = 4n + 6

3x3 = 4n + 8

4x3 = 4n + 10

5x3 = 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 5x3 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

1x1 = 4n

2x1 = 4n + 2

3x1 = 4n + 4

4x1 = 4n + 6

5x1 = 4n + 8

Formulas for nx2

1x1 = 4n + 2

2x1 = 4n + 4

3x1 = 4n + 6

4x1 = 4n + 8

5x1 = 4n + 10

Formulas for nx3

1x1 = 4n + 4

2x1 = 4n + 6

3x1 = 4n + 8

4x1 = 4n + 10

5x1 = 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 2x2 to 10x15. 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.

Created by Syed Islam, 11Q