# Investigate Borders - a fencing problem.

Borders

Borders

Aim

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.

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 1x1 square, then 2x1 and so on up to 5x1. 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 1x2 square, then 2x2, and so on up to 5x2, then 1x3, 2x3, and so on up to 5x3. 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: 1x1

Table of results for Borders of square: 1x1

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 1x1):

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: 2x1

Table of results for Borders of square: 2x1

Formula to find the number of squares needed for each border (for square 2x1):

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: 3x1

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