Practice Maths

Describing Patterns Algebraically

Key Ideas

Key Terms

Arithmetic sequence
A sequence where consecutive terms differ by a constant amount called the common difference.
Common difference
The fixed amount added (or subtracted) between consecutive terms in an arithmetic sequence, denoted d.
nth term
A formula Tn = a + (n − 1)d that gives the value of any term by its position n, where a is the first term.
Linear pattern
A pattern where values increase or decrease by a constant amount; produces a straight-line graph when plotted.
General rule
An algebraic expression (e.g. Tn = 3n + 1) that describes every term using n as the position number.

Connecting Sequences to Linear Functions

The nth term formula Tn = dn + (a − d) is a linear function of n. The common difference d is the gradient, and (a − d) is the y-intercept.

Hot Tip For linear sequences, the common difference d is the coefficient of n. Use the shortcut: Tn = dn + (a − d), where a is the first term. For 3, 7, 11, 15…: d = 4, a = 3, so Tn = 4n + (3 − 4) = 4n − 1. Check: T1 = 3 ✓, T2 = 7 ✓.

Worked Example

Question: Find the nth term rule for the sequence 5, 9, 13, 17…

Common difference: d = 9 − 5 = 4.
First term: a = 5.
Tn = 4n + (5 − 4) = 4n + 1.
Check: T1 = 5 ✓, T3 = 13 ✓

Is 89 a term in this sequence?
Set 4n + 1 = 89 → 4n = 88 → n = 22. Yes, 89 is the 22nd term.

Patterns and Tables of Values

Many real-world situations can be described as patterns. A linear pattern is one where the values increase (or decrease) by the same amount each time — a constant difference. For example, if you earn $15 per hour, after 1 hour you have $15, after 2 hours $30, after 3 hours $45 — the pattern goes up by $15 each time.

A table of values is a neat way to organise a pattern. The input (usually called x or n, representing the step number) goes in one row or column, and the corresponding output value (usually called y) goes in the other.

To check if a pattern is linear, calculate the differences between consecutive output values. If all the differences are the same (constant difference), it is linear. If the differences are not constant, the pattern is not linear.

Writing a Rule: y = mx + b

A linear pattern can always be described by a rule of the form y = mx + b, where:

  • m is the constant difference (how much y changes each time x increases by 1) — this is the gradient or rate of change.
  • b is the starting value when x = 0 — this is the y-intercept.

To find m: calculate the constant difference between consecutive y values. To find b: work backwards or substitute a known x and y value into the rule once you know m.

Example: The table shows x = 1, 2, 3, 4 and y = 7, 10, 13, 16. The difference between each y value is +3, so m = 3. When x = 1, y = 7: 7 = 3(1) + b, so b = 4. Rule: y = 3x + 4.

Check: when x = 4, y = 3(4) + 4 = 16. ✓

Predicting Values Using the Rule

Once you have the algebraic rule, you can predict any value in the pattern — not just the next one or two, but any step, including very large ones. This is the power of algebra: instead of extending a table step by step for 100 rows, just substitute x = 100 into the rule.

Example (using y = 3x + 4): What is the 50th value? y = 3(50) + 4 = 150 + 4 = 154. Easy!

You can also work backwards: if y = 40, what is x? 40 = 3x + 4, so 3x = 36, x = 12. That means the value 40 is the 12th term.

Connecting Patterns to Graphs

When you plot a linear pattern as a graph (x on the horizontal axis, y on the vertical axis), you get a straight line. The gradient m tells you how steeply the line rises or falls, and the y-intercept b tells you where the line crosses the vertical axis.

A positive gradient means the pattern increases (line goes up left to right). A negative gradient means the pattern decreases (line goes down left to right). A gradient of zero means the pattern is constant (a horizontal line).

Seeing both the table and the graph together gives a complete picture of the pattern — the table shows exact values, and the graph shows the overall shape and trend.

Non-Linear Patterns — Recognising the Difference

Not all patterns are linear. If the differences between consecutive y values are not constant — for example, if they double each time (1, 2, 4, 8, 16, ...) — the pattern is non-linear. When plotted, non-linear patterns form curves, not straight lines. In Year 8, the focus is on identifying and working with linear patterns, but it is useful to recognise when a pattern is not linear so you know the linear rule won't work.

Key tip: To find the rule for a linear pattern, first find m (the constant difference between y values) and then find b by substituting any known point into y = mx + b and solving for b. Don't just assume b equals the first y value — that is only true if the first x value is 1 and m happens to match up. Always substitute and solve properly to be sure.

Mastery Practice

  1. Find the common difference and write the next 3 terms for each sequence. Fluency

    1. 2, 5, 8, 11, …
    2. 10, 7, 4, 1, …
    3. 3, 8, 13, 18, …
    4. 20, 16, 12, 8, …
    5. 1, 1.5, 2, 2.5, …
    6. −3, 0, 3, 6, …
    7. 100, 93, 86, 79, …
    8. 4, 4.5, 5, 5.5, …

  2. Write the nth term formula Tn for each sequence. Fluency

    1. 3, 7, 11, 15, …
    2. 5, 8, 11, 14, …
    3. 10, 8, 6, 4, …
    4. 1, 4, 7, 10, …
    5. 6, 11, 16, 21, …
    6. 20, 17, 14, 11, …
    7. 2, 2.5, 3, 3.5, …
    8. −1, 2, 5, 8, …

  3. Use the nth term formula to find the specified terms. Fluency

    1. Tn = 3n + 1: find T20, T50, T100
    2. Tn = 5n − 2: find T20, T50, T100
    3. Tn = −2n + 50: find T10, T20, T25
    4. Sequence 4, 9, 14, 19…: find the 20th, 50th and 100th terms
    5. Sequence 7, 11, 15, 19…: find the 20th, 50th and 100th terms
    6. Sequence 200, 195, 190, 185…: find the 20th, 40th and 50th terms
    7. Tn = 2n + 0.5: find T10, T50, T100
    8. Sequence −5, −1, 3, 7…: find the 15th, 30th and 50th terms

  4. Set up and solve an equation to determine whether each value is a term in the given sequence. If it is, state which term. Understanding

    1. Is 47 a term in 5, 8, 11, 14, …?
    2. Is 100 a term in 3, 7, 11, 15, …?
    3. Is 85 a term in 4, 9, 14, 19, …?
    4. Is 200 a term in 6, 11, 16, 21, …?
    5. Is 55 a term in 2, 6, 10, 14, …?
    6. Is −40 a term in 10, 7, 4, 1, …?
    7. Is 1000 a term in 7, 14, 21, 28, …?
    8. Is 72 a term in 8, 15, 22, 29, …?

  5. For each shape pattern described, complete the table, find the nth term rule and answer the question. Understanding

    1. Matchstick squares in a row: 1 square needs 4 sticks, 2 squares need 7 sticks, 3 squares need 10 sticks.
      1. Write the sequence of matchstick counts for 1 to 5 squares.
      2. Find the nth term rule.
      3. How many sticks for 20 squares?
    2. Triangles in a row sharing sides: 1 triangle needs 3 sticks, 2 need 5 sticks, 3 need 7 sticks.
      1. Find the nth term rule.
      2. How many triangles can be made with 31 sticks?
    3. Dots arranged in an L-shape: stage 1 has 3 dots, stage 2 has 5 dots, stage 3 has 7 dots.
      1. Find the nth term rule.
      2. Which stage has 25 dots?
    4. Tiles in a plus (cross) pattern: stage 1 has 5 tiles, stage 2 has 9 tiles, stage 3 has 13 tiles.
      1. Find the nth term rule.
      2. Which stage has 49 tiles?
    5. Hexagons sharing sides: 1 hexagon needs 6 sticks, 2 need 11 sticks, 3 need 16 sticks.
      1. Find the nth term rule.
      2. How many sticks for 10 hexagons?
    6. Square spiral: stage 1 has 1 square, stage 2 has 3 squares, stage 3 has 5 squares.
      1. Find the nth term rule.
      2. Is 100 squares possible? Why or why not?

  6. Use your sequence skills to solve these problems. Problem Solving

    1. A cinema has 20 seats in the first row, 24 seats in the second row, 28 seats in the third row, and so on.
      1. Write the nth term formula for the number of seats in row n.
      2. How many seats are in the 15th row?
      3. A theatre has 20 rows. Write an expression for the total number of seats. (Hint: add T1 + T2 + … + T20.)
    2. A new swimming pool is being filled. At the end of hour 1, it contains 800 L. Each hour, 600 L more is added.
      1. Find the nth term rule for the volume after n hours.
      2. After how many hours will the pool contain 5000 L?
      3. The pool capacity is 8000 L. Will the pool be full after 12 hours?
    3. Two arithmetic sequences are given: Sequence A: 3, 7, 11, 15… and Sequence B: 5, 8, 11, 14…
      1. Write the nth term rule for each.
      2. Find the value(s) that appear in both sequences for n ≤ 20.
      3. Explain whether the sequences will keep sharing terms.
    4. A stack of cans forms a triangular display. Row 1 (bottom) has 10 cans, row 2 has 9, and so on up to row 10 with 1 can.
      1. Write the rule for the number of cans in row n (counting from the bottom).
      2. Which row has 6 cans?
      3. Find the total number of cans in the display.
  7. Linear vs Non-linear. Identifying the type of pattern is a key skill before finding a rule.

    Classify each pattern as linear or non-linear. Justify your answer, and find the nth term rule for any linear patterns. Problem Solving

    1. 1, 4, 9, 16, 25, …
    2. 3, 5, 7, 9, 11, …
    3. 2, 4, 8, 16, 32, …
    4. 10, 7, 4, 1, −2, …
    5. 0, 3, 8, 15, 24, … (differences: 3, 5, 7, 9)
    6. 100, 97, 94, 91, …
  8. Patterns in Context. Two companies track their weekly sales.

    Company A starts with 12 sales in week 1 and increases by 5 sales per week. Company B starts with 32 sales in week 1 and increases by 2 sales per week. Problem Solving

    1. Write the nth term rule for Company A’s weekly sales.
    2. Write the nth term rule for Company B’s weekly sales.
    3. In which week do the two companies have the same sales? Show your algebraic working.
    4. After 10 weeks, which company has more total sales? (Add T1 through T10 for each, or note that you can use the sum formula if known.)
  9. Fencing Problem. A farmer uses fence panels to build a row of square paddocks.

    Each square paddock uses 4 fence panels. When paddocks share a side, they share a panel. Problem Solving

    1. Complete the table for the number of fence panels needed:
      Paddocks (n)12345
      Panels     
    2. Find the nth term rule for the number of panels.
    3. How many panels are needed for 25 paddocks?
    4. The farmer has 76 panels. What is the maximum number of paddocks he can build?
  10. Working Backwards. Sometimes you know a term value and need to find the position.

    Solve each reverse sequence problem. Problem Solving

    1. A sequence has nth term rule Tn = 7n − 3. Which term has value 375?
    2. A sequence starts at 8 and increases by 6 each term. Which term first exceeds 200?
    3. A sequence has first term 150 and decreases by 4 each term. Which is the last positive term? What is its value?
    4. Two sequences share the rule structure Tn = an + b. Sequence P has T1 = 5 and T4 = 17. Sequence Q has T2 = 10 and T5 = 19. Find the nth term rule for each, then find the first term they share in common (if any, for n ≤ 30).
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