Module 3 - Investigation Structure
TL;DR
Finding the general rule is the key step in the investigation.
You just have to write it down; no need to show any working.
Stating the General Rule is where you put all the skills from Module 2 to work.
The command term for this point is to Write Down which means you do not need to give any working here to show how you got the general term (that is assessed in criteria J, justify).
This is marked against criteria D, alongside describing the patterns.
Usually there are 4 marks in total for criteria D. You get 3 of those marks if you write down a correct general rule for the sequence in terms of \(n\).
You get the full 4 marks if you have a correct general term AND have described a valid pattern.
There are 2 marks available if you attempt to get the general rule, but make a small mistake.
If you get the general rule wrong, then you will not be able to get many marks in the other sections of the unstructured investigation.
It is imperative that you are confident in quickly and accurately finding the general rule for linear, quadratic and geometric sequences, and that you are happy with hidden sequences too.
Example
Write down the general rule for \(C\) in terms of \(n\).
| Stage (\(n\)) | Cubes (\(C\)) |
|---|---|
| 1 | 12 |
| 2 | 19 |
| 3 | 26 |
| 4 | 33 |
| 5 | 40 |
| 6 | 47 |
| 7 | 54 |
-
The values for \(C\) follow a linear pattern. They increase by \(7\) each time.
$$C=7n+5$$
Example
Write down the general rule for \(C\) in terms of \(n\).
| Stage (\(n\)) | Count (\(C\)) |
|---|---|
| 1 | 100 |
| 2 | 93 |
| 3 | 86 |
| 4 | 79 |
| 5 | 72 |
| 6 | 65 |
| 7 | 58 |
-
The values for \(C\) follow a linear pattern. They decrease by \(7\) each time.
$$C=-7n+107$$
Example
Write down the general rule for \(T\) in terms of \(n\).
| Stage (\(n\)) | Tiles (\(T\)) |
|---|---|
| 1 | 4 |
| 2 | 7 |
| 3 | 12 |
| 4 | 19 |
| 5 | 28 |
| 6 | 39 |
| 7 | 52 |
-
The values for \(T\) follow a quadratic pattern. The second difference is \(2\).
$$T=n^{2}+3$$
Example
Write down the general rule for \(S\) in terms of \(n\).
| Stage (\(n\)) | Squares (\(S\)) |
|---|---|
| 1 | 5 |
| 2 | 10 |
| 3 | 20 |
| 4 | 40 |
| 5 | 80 |
| 6 | 160 |
| 7 | 320 |
-
The values for \(S\) follow a geometric pattern. Each term is the previous term multiplied by \(2\).
$$S=5\times \left(2\right)^{n-1}$$
Example
Write down the general rule for \(P\) in terms of \(n\).
| Stage (\(n\)) | Pieces (\(P\)) |
|---|---|
| 1 | 0 |
| 2 | 3 |
| 3 | 8 |
| 4 | 15 |
| 5 | 24 |
| 6 | 35 |
| 7 | 48 |
-
The values for \(P\) follow a quadratic pattern. The second difference is \(2\).
$$P=n^{2}-1$$
Example
Write down the general rule for \(C\) in terms of \(n\).
| Stage (\(n\)) | Circles (\(C\)) |
|---|---|
| 1 | 10 |
| 2 | 12 |
| 3 | 16 |
| 4 | 22 |
| 5 | 30 |
| 6 | 40 |
| 7 | 52 |
-
The values for \(C\) follow a quadratic pattern. The second difference is \(2\).
$$C=n^{2}-n+10$$
Example
Write down the general rule for \(L\) in terms of \(n\).
| Stage (\(n\)) | Length (\(L\)) |
|---|---|
| 1 | 48 |
| 2 | 24 |
| 3 | 12 |
| 4 | 6 |
| 5 | 3 |
| 6 | 1.5 |
| 7 | 0.75 |
-
The values for \(L\) follow a geometric pattern. Each term is the previous term multiplied by \(\frac{1}{2}\).
$$L=48\times\left(\frac{1}{2}\right)^{n-1}$$
Example
Write down the general rule for \(A\) in terms of \(n\).
| Stage (\(n\)) | Area (\(A\)) |
|---|---|
| 1 | \(\sqrt{7}\) |
| 2 | \(\sqrt{11}\) |
| 3 | \(\sqrt{15}\) |
| 4 | \(\sqrt{19}\) |
| 5 | \(\sqrt{23}\) |
| 6 | \(\sqrt{27}\) |
| 7 | \(\sqrt{31}\) |
-
The values for \(A\) inside the square root follow a linear pattern. They increase by \(4\) each time.
$$A=\sqrt{4n+3}$$
Example
Write down the general rule for \(M\) in terms of \(n\).
| Stage (\(n\)) | Midpoint (\(M\)) |
|---|---|
| 1 | \(\left(2,5\right)\) |
| 2 | \(\left(2,8\right)\) |
| 3 | \(\left(2,15\right)\) |
| 4 | \(\left(2,26\right)\) |
| 5 | \(\left(2,41\right)\) |
| 6 | \(\left(2,60\right)\) |
| 7 | \(\left(2,83\right)\) |
-
The first value in the coordinate for \(M\) is always \(2\).
The second value in the coordinate for \(M\) follows a quadratic pattern. The second difference is \(4\).
$$M=\left(2,2n^{2}-3n+6\right)$$
Example
Write down the general rule for \(R\) in terms of \(n\).
| Stage (\(n\)) | Ratio (\(R\)) |
|---|---|
| 1 | \(\frac{15}{3}\) |
| 2 | \(\frac{13}{9}\) |
| 3 | \(\frac{11}{27}\) |
| 4 | \(\frac{9}{81}\) |
| 5 | \(\frac{7}{243}\) |
| 6 | \(\frac{5}{729}\) |
| 7 | \(\frac{3}{2187}\) |
-
The numerator of \(R\) follows a linear pattern. They decrease by \(2\) each time.
The denominator of \(R\) follows a geometric pattern. Each term is the previous term multiplied by \(3\).
$$R=\frac{-2n+17}{3\times 3^{n-1}}$$
General Rule
| Stage (\(n\)) |
|---|