Patterns and Algebra - Investigating Patterns and Sequences

A sequence is an ordered list of numbers or objects called 'terms'. For example, in the sequence $3, 6, 9, 12, \dots$, each number is a term, and they follow a specific rule of adding $3$ to the previous number.

The position of a term is represented by $n$. In a table of values, $n$ usually represents the input (stage number) and $T$ represents the output (the value at that stage). Visually, if you have a pattern of shapes, such as a row of growing triangles, $n$ is the figure number and $T$ is the count of individual pieces in that figure.

An arithmetic sequence is a pattern where the difference between any two consecutive terms is constant. This is called the common difference ($d$). If you plot these terms as coordinates $(n, T)$ on a grid, the points will align perfectly on a straight diagonal line, showing a linear relationship.

The $n^{th}$ term formula is an algebraic expression that represents the rule for a sequence. For linear sequences, it takes the form $T_{n} = dn + c$, where $d$ is the common difference. For example, if $T_{n} = 5n - 2$, the $10^{th}$ term is $5(10) - 2 = 48$.

Function Machines provide a visual way to process rules. Imagine a box where an 'Input' number $n$ enters, a mathematical operation is performed (like $\times 3$), and an 'Output' $T$ emerges. This helps visualize how the position relates to the term value through a specific process.

Geometric growth can be seen in patterns where shapes expand in two dimensions. For example, a square that increases its side length by $1$ unit at each stage will have an area sequence of $1, 4, 9, 16, \dots$ ($n^{2}$). Visually, these patterns appear to 'accelerate' or curve upwards on a graph rather than forming a straight line.

To solve for the rule of a sequence, find the common difference ($d$) first. Then, find the value that would exist at 'Position 0' by subtracting the difference from the first term. If the first term is $10$ and $d=4$, the 'zeroth' term is $6$, making the rule $T = 4n + 6$.