Algebra: Sequences and Graphs

Sequences and graphs are two of the most important ways mathematics describes patterns and relationships. At KS3 you will find rules for sequences, plot and interpret straight-line and quadratic graphs, and understand what graphs tell us about the relationship between two quantities.

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Sequences

A sequence is an ordered list of numbers. Each number is called a term.

Arithmetic (Linear) Sequences

The difference between consecutive terms is constant — called the common difference (d).

Example: 3, 7, 11, 15, 19, … → d = +4 (add 4 each time)

nth term formula: nth term = a + (n − 1)d, often simplified to dn + c

• d = common difference
• c = (first term) − d

Example: Find the nth term of 5, 8, 11, 14, … d = 3; c = 5 − 3 = 2 → nth term = 3n + 2

Check: n=1 → 3+2=5 ✓; n=5 → 15+2=17 ✓

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Geometric Sequences

Each term is multiplied by a constant — the common ratio (r).

Example: 2, 6, 18, 54, … → r = 3 (multiply by 3 each time)

nth term = a × r^(n−1), where a = first term.

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Recognising Other Sequences

Sequence	Description
1, 4, 9, 16, 25, …	Square numbers (n²)
1, 8, 27, 64, …	Cube numbers (n³)
1, 1, 2, 3, 5, 8, 13, …	Fibonacci sequence
2, 3, 5, 7, 11, 13, …	Prime numbers

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Coordinates

Coordinates describe a position using an x-value (horizontal) and a y-value (vertical), written as (x, y).

At KS3 you work in all four quadrants: top-right is Q1 (+x, +y), top-left is Q2 (−x, +y), bottom-left is Q3 (−x, −y), and bottom-right is Q4 (+x, −y).

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Linear Graphs (Straight Lines)

Equation y = mx + c

Variable	Meaning
m	gradient (steepness) = rise ÷ run = Δy/Δx
c	y-intercept (where the line crosses the y-axis)

Gradient sign: positive → line slopes upward; negative → line slopes downward.

Example: y = 3x − 2 has gradient 3 and y-intercept −2.

Plotting a line: choose three x values (e.g. 0, 1, 2), calculate y, then join the points.