Algebra - Investigating Patterns and Sequences

A sequence is an ordered list of numbers where each number is called a 'term'. For example, in the sequence $2, 4, 6, 8, \dots$, the number $2$ is the 1st term (position $n=1$), and $4$ is the 2nd term (position $n=2$). Visually, this can be represented as a series of steps or blocks that increase in size following a specific rule.

Arithmetic Sequences (Linear Patterns) are sequences where the difference between any two consecutive terms is constant. This constant is called the common difference ($d$). If you plot these terms on a graph with position $n$ on the x-axis and the term value on the y-axis, the points will form a perfectly straight line, showing a linear relationship.

The Term-to-Term Rule describes how to get from one term to the next. For example, in the sequence $5, 8, 11, \dots$, the term-to-term rule is 'add $3$'. This represents a constant growth where the same visual element (like a row of dots or a matchstick) is added at every single step.

The Position-to-Term Rule (or $n^{th}$ term rule) is an algebraic expression that relates the position of a term ($n$) to its value. This allows you to calculate the value of any term (like the 100th term) without knowing the one before it. In the form $u_n = dn + c$, $d$ is the common difference and $c$ is the value that would exist at position $0$.

Finding the Common Difference ($d$) is done by subtracting any term from the one that follows it: $d = u_{n+1} - u_n$. If $d$ is positive, the sequence is increasing and the visual pattern expands; if $d$ is negative, the sequence is decreasing and the visual pattern shrinks.

The Constant $c$ in the formula $u_n = dn + c$ represents the 'zero term' ($u_0$). You can calculate it by subtracting the common difference from the first term: $c = u_1 - d$. Visually, if a pattern of tiles starts with 5 tiles and adds 2 each time, the $c$ value represents the 'base' or starting amount before the first growth step.

Geometric Patterns involve visual shapes (like squares, triangles, or dots) that grow. To solve these, you first translate the shapes into a numerical sequence. For instance, a sequence of houses made of matchsticks where the first house uses $6$ sticks and each additional house shares a wall and adds $5$ more sticks creates the sequence $6, 11, 16, \dots$.