Can You Use the nth Term for This 11+ Maths Visual Pattern Question?

Hi everyone, welcome back to another St Olab’s question by Singapore Maps Academy! My name is Mohi, and today, I’ll be taking you through a really interesting patterns question. You might not initially think it’s about the nth term, but understanding the nth term can really help you with solving it.

Understanding the Problem

The problem states that a shape is made from 2016 small squares. By continuing the pattern shown in the diagram (not provided here), the objective is to determine the length in centimetres of the perimeter of the whole shape. Each small square has a side length of one centimetre.

Initial Thoughts

When tackling perimeter questions, it’s common to think about tracing the outside length of the shape:

“What is the length in centimetres of the perimeter of the whole shape? Now that can seem a bit tricky because you’re looking at perimeter.”

To solve this effectively, let’s break it down with a simpler example and gradually build up.

Breaking Down the Shape

One Column Example

Let’s start with just one column of squares:

 IMAGE-1-HERE

Calculating the Perimeter

For column one, each small square has a side length of one centimetre. So, the perimeter of column one would be:

 IMAGE-2-HERE

123456 (Counted for the total perimeter of one column)

“For column one, the perimeter is going to be six. Does that make sense?”

Adding More Columns

Now, what if we add a second column next to the first?

 IMAGE-3-HERE

1. We start with the first column’s perimeter of 6.
2. We need to adjust because we add and remove certain sides due to the adjacent columns.

Hence, with two columns, the perimeter changes:

1234-5678, 910

Simpler form:This nested structure shows:

First column: 6 cmSecond column: +4 more sides

This means, for two columns, the perimeter sums up to:

6 + 4 = 10 cm

Third Column

Extending the concept further, with three columns:

Additional 4 more sides

= 10 cm (Two columns) + 4 = 14 cm

Recognizing the Pattern

Clearly, with each new column, we continue adding 4 to the perimeter:

“So I can see that I’m just going to be adding four each time. And because of that, I know. Therefore, the nth term for this pattern of 610 14 is going to be four n because it goes up in fours.”

Creating the nth Term Formula

Let’s derive the nth term:

Formula Structure

Given the pattern with a common difference of 4 (basic arithmetic progression), our nth term formula structure begins with 4n.

General form: 4n + c

Calculating the Constant (c)

To find c, observe that for n = 1 (one column), the perimeter is 6.

Therefore:

4 * 1 + c = 6c = 2

Hence the formula:

nth term = 4n + 2

Applying the Formula to the Original Problem

“Now, it says that we have a shape made of 2016 small squares. Now, if each column has two squares, how am I going to do this?”

Calculating the Number of Columns

Each column consists of two squares. Thus:

Total columns = Total squares / Squares per columnTotal columns = 2016 / 2Total columns = 1008

Determining the Perimeter

Now, applying our derived nth term formula:

For n = 1008:Perimeter = 4(1008) + 2Perimeter = 4032 + 2Perimeter = 4034 cm

Conclusion

The shape made of 2016 small squares has a perimeter of:

> "4034 is the perimeter. Centimetres is the perimeter of the shape that has 20 2016 squares."

Understanding visual patterns and deriving nth term formulas can simplify and solve seemingly complex perimeter questions. I hope you found this helpful. Let me know your thoughts in the comments below. Take care and see you in the next video!