Notice what changes, classify the rule, and extend number and diagram patterns with confidence.
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\[\text{Notice}\rightarrow\text{Describe}\rightarrow\text{Extend}\rightarrow\text{Check}\]
Introduction
A sequence is more than a list of numbers: its terms are connected by a rule. In this lesson you will learn to identify that relationship before extending the pattern.
Which question should you ask first?
\[7,\ 11,\ 15,\ 19,\ 23,\ldots\]
Topic 1 · Pattern language
A term is a value in a sequence. Its position tells where it occurs. Here, 15 is the third term and 23 is the fifth term.
\[T_1=7\qquad T_3=15\qquad T_5=23\]
Topic 1 · Pattern language
We use (T_n) to name the term at position (n). The subscript is a position label; it is not multiplication.
\[\begin{array}{c|ccccc}\text{Position}&1&2&3&4&5\\\hline\text{Term}&7&11&15&19&23\end{array}\]
Topic 1 · Pattern language
The top row records position; the bottom row records the matching term. This becomes especially useful when diagram patterns and formulas are introduced.
\[\text{Pattern}=\text{ordered objects or numbers following a rule}\]
Topic 1 · Pattern language
Useful descriptions name the operation and amount: “add 4 each time”, “multiply by 3”, or “square the position”. Avoid vague descriptions such as “the numbers go up”.
\[\boxed{\text{position }n\longleftrightarrow\text{ term }T_n}\]
Topic 1 · Checkpoint
You can identify a term, state its position, read (T_n), and organise a pattern in a position–term table.
\[12,\ 17,\ 22,\ 27,\ldots\qquad +5,+5,+5\]
Topic 2 · Constant difference
Subtract consecutive terms. If every difference is equal, the sequence has a constant difference. Here the difference is (+5).
\[30,\ 24,\ 18,\ 12,\ldots\qquad -6,-6,-6\]
Topic 2 · Constant difference
A decreasing arithmetic pattern still has a rule. Subtracting 6 each time means the constant difference is (-6).
\[\ldots,\ 14,\ 21,\ 28,\ 35,\ldots\]
Topic 2 · Constant difference
Forward movement adds 7, so backward movement subtracts 7. The two terms before 14 are 7 and 0.
\[-8,\ -3,\ 2,\ 7,\boxed{12},\boxed{17}\]
Topic 2 · Constant difference
Check each pair: (-3-(-8)=5), (2-(-3)=5), and (7-2=5). Continue by adding 5.
What is the constant difference?
\[\boxed{\text{same difference}\Rightarrow\text{arithmetic pattern}}\]
Topic 2 · Checkpoint
You can calculate consecutive differences, continue increasing or decreasing arithmetic patterns, and extend them backwards.
\[2,\ 6,\ 18,\ 54,\ldots\qquad\times3\]
Topic 3 · Constant multiplier
Compare each term with the previous term. Here every term is three times the one before it, so the constant multiplier is 3.
\[96,\ 48,\ 24,\ 12,\ldots\qquad\div2\]
Topic 3 · Constant multiplier
Dividing by 2 is the same as multiplying by ( frac12). The next terms are 6 and 3.
\[3,6,9,12\quad\neq\quad3,6,12,24\]
Topic 3 · Constant multiplier
The first sequence adds 3. The second multiplies by 2. Always test the proposed rule against every consecutive pair.
\[5,\ 25,\ 125,\ 625=5^1,5^2,5^3,5^4\]
Topic 3 · Constant multiplier
Each term is multiplied by 5. The same pattern can be described using powers of 5.
What is the next term?
\[\boxed{\text{same multiplier}\Rightarrow\text{geometric pattern}}\]
Topic 3 · Checkpoint
You can recognise multiplication and division patterns, express division as a fractional multiplier, and distinguish them from arithmetic patterns.
\[1,\ 4,\ 9,\ 16,\ 25,\ldots\]
Topic 4 · Other patterns
These terms are (1^2,2^2,3^2,4^2,5^2). The next terms are (6^2=36) and (7^2=49). The differences change: 3, 5, 7, 9, ...
\[2,\ 5,\ 4,\ 10,\ 8,\ 20,\ldots\]
Topic 4 · Other patterns
Odd positions give 2, 4, 8, ... and even positions give 5, 10, 20, .... Each linked sequence doubles. The next terms are 16 and 40.
\[3,\ 6,\ 11,\ 18,\ 27,\ldots\qquad +3,+5,+7,+9\]
Topic 4 · Other patterns
The differences are consecutive odd numbers. The next difference is 11, so the next term is (27+11=38).
\[\frac12,\ \frac23,\ \frac34,\ \frac45,\ldots\]
Topic 4 · Other patterns
The numerator and denominator each increase by 1. The next two terms are ( frac56) and ( frac67). Not every rule compares whole terms by one operation.
\[\boxed{\text{Try squares, powers, alternating parts, and changing differences}}\]
Topic 4 · Checkpoint
You can investigate patterns beyond a constant difference or multiplier and explain the structure you found.
1 2 3 4
Topic 5 · Diagrams and tables
Record both quantities. In this simple pattern, the number of tiles equals the figure number.
\[\begin{array}{c|ccccc}n&1&2&3&4&5\\\hline T_n&4&7&10&13&16\end{array}\]
Topic 5 · Diagrams and tables
If each new figure adds 3 tiles, the table displays a constant-difference pattern. Figure 6 therefore has 19 tiles.
\[6,\ 10,\ 14,\ 18,\boxed{22}\]
Topic 5 · Diagrams and tables
The tile count increases by 4 for each new figure. Figure 5 has 22 tiles.
\[\text{diagram}\longleftrightarrow\text{table}\longleftrightarrow\text{words}\]
Topic 5 · Diagrams and tables
A diagram shows the structure, a table records the values, and words explain the change. Moving between them makes the rule easier to defend.
\[\boxed{\text{figure number}\longleftrightarrow\text{number of objects}}\]
Topic 5 · Checkpoint
You can connect a figure to its tile count, make a position–value table, describe the growth, and predict the next figure.
\[\text{Look}\rightarrow\text{test}\rightarrow\text{extend}\rightarrow\text{check}\]
Topic 6 · Investigate and check
\[11,\ 15,\ 19,\ 23,\boxed{27},\boxed{31},\boxed{35}\]
Topic 6 · Investigate and check
The pattern adds 4 each time. A complete answer gives the next terms and the rule in words.
\[2,\ 6,\ 12,\ 20,\ 30,\ldots\qquad +4,+6,+8,+10\]
Topic 6 · Investigate and check
The differences are consecutive even numbers. Checking all pairs prevents the incorrect claim that the pattern has one constant difference.
What difference comes next?
\[\text{A short list may fit more than one rule}\]
Topic 6 · Investigate and check
With only a few terms, several possible rules may exist. Use the context and explain why your rule produces every displayed term.
\[\boxed{\text{A good rule reproduces every given term}}\]
Topic 6 · Checkpoint
You can classify unfamiliar patterns, extend them, state a precise rule, and verify that the rule works throughout the sequence.
\[\begin{aligned}\text{same difference}&\Rightarrow\text{arithmetic}\\\text{same multiplier}&\Rightarrow\text{geometric}\\\text{other structure}&\Rightarrow\text{squares, powers, alternating parts}\end{aligned}\]
Summary
Start by identifying terms and positions. Compare consecutive terms, classify the structure, extend it, and check your rule against all the information.
You are ready to represent patterns with general rules.