Numbers Patterns

Categorising Sequences: Differences and Ratios

To identify how a pattern behaves, calculate the differences or ratios between consecutive terms:

• Non-linear patterns are any patterns without a constant first difference; quadratic and exponential patterns are both non-linear.

Linear Patterns and the General Term

A linear sequence changes by adding or subtracting the same value, $d$, at each step.

When counting starts at term 1 ($n = 1$), the general term is:

• $a$ is the first term ($T_1$).
• $d$ is the common difference ($T_2 - T_1$).
• $n$ is the term position ($n \in \mathbb{N}$).

For $5, 8, 11, 14, \dots$, $a = 5$ and $d = 3$:

In a real-world linear pattern, the starting amount is the Stage 0 value and the fixed change each step is $d$:

Saving €15 at the start and €6 a week gives $T_n = 15 + 6n$ after $n$ weeks. To find when the total is €81, solve $15 + 6n = 81 \implies 6n = 66 \implies n = 11$ weeks.

Quadratic Patterns and Finding Tn

A quadratic pattern has a constant, non-zero second difference and general term:

To find the coefficients:

1. Find $a$: The second difference equals $2a$, so $a = \frac{\text{second difference}}{2}$.
2. Form equations: Substitute $n = 1$ and $n = 2$ into $T_n = an^2 + bn + c$ using the given values of $T_1$ and $T_2$.
3. Solve for $b$ and $c$: Subtract the two equations to eliminate $c$, find $b$, and substitute back to find $c$.

A negative second difference gives a negative $a$, so the terms eventually fall and the curve turns downwards. Finish by checking your formula gives the correct $T_3$, and be ready to state the rule in words.

Visual Patterns and Discrete Graphs

Exam questions often present spatial patterns, such as tile grids or matchsticks.

For joined matchstick squares, count the shared sides: an initial square needs $4$ sticks, and each extra square adds $3$ sticks:

For a linear pattern beginning at Stage 0, use $T_n = dn + T_0$. Here, working backwards gives a Stage 0 base of $1$ stick, leading directly to $3n + 1$.

When graphing sequences, plot position $n$ on the horizontal axis and term value $T_n$ on the vertical axis as discrete, unconnected points. Do not join the points with a line, because term positions are whole numbers.

Key terms

Term

An individual number or member at a specific position in a sequence, written as $T_n$.

Common Difference (d)

The constant amount added to or subtracted from each term to get the next term in a linear sequence.

Second Difference

The difference between consecutive first differences; it is always constant and non-zero in a quadratic sequence.

Common Ratio

The constant factor by which each term is multiplied to produce the next term in an exponential sequence.

General Term (Tn)

An algebraic expression that produces the value of any term in a sequence from its position number $n$.