GCSE · Maths · AQA · Spec 8300
Generating sequences from term-to-term or position-to-term rules
Start at 7 and keep taking away 3. Four steps later you're below zero, and you can say exactly where. That's a sequence: one rule, used over and over.
Patterns are sequences too
Matchstick squares in a row. Pattern 1 is one square: 4 matchsticks. Pattern 2 is two squares side by side, sharing a stick: 7 matchsticks. Pattern 3 is three squares in a row: 10 matchsticks.
The row keeps growing one square at a time. How many matchsticks does Pattern 10 need?
WHAT YOU'VE LEARNED
A quick recap of today's lesson.
Walk from the term before, or jump straight to any position.
What you need to know
- A term-to-term rule gets the next term from the term before, so it needs a first term to start from.
- A position-to-term rule gets a term from its position number, so you can find any term directly.
- Before you use a rule, check what goes in: the previous term, or the position number.
- In an arithmetic sequence the term-to-term rule is the common difference. It can be positive or negative (the sequence goes up or down) and can be any value, including decimals.
- In a growing pattern the piece counts are the terms. If each new diagram adds the same number of pieces, adding that number is the term-to-term rule.
The big picture
A sequence is a list of numbers in order, and each number is a term. A term-to-term rule makes each term from the one before, so you walk along from a first term. A position-to-term rule makes each term from its position number, so you can jump straight to any term. In an arithmetic sequence you add the same amount every time. That common difference is the term-to-term rule, and it can be positive, negative or not a whole number. Growing patterns work the same way: the piece counts are the terms.
Key points
Worked example
Problem
A sequence has the position-to-term rule 'multiply the position by 4, then subtract 1'. (a) Write down the first three terms. (b) Find the 20th term. (c) Write down the term-to-term rule.
⚠ Watch out
Feeding the previous term into a position-to-term rule. For 'multiply the position by 10, then subtract 1', the 2nd term is 10 × 2 − 1 = 19, not 10 × 9 − 1 = 89, which uses the 1st term, 9. Ask first: what goes in?
Memory hook
Term-to-term = WALK: one step at a time from where you are. Position-to-term = JUMP: straight to any spot. Before you move, ask what the rule is fed.
Check yourself
Rule: 'multiply the position by 5, then subtract 3'. Write the first three terms, the term-to-term rule and the 30th term. (Answers: 2, 7, 12; add 5; 147.)
Flashcards
(9)What does a term-to-term rule use to make the next term?
What does a position-to-term rule use to make a term?
Why doesn't 'add 3' on its own pin down one sequence?
In an arithmetic sequence, what is the term-to-term rule?
Can the common difference be negative?
Does the common difference have to be a whole number?
How do you get the terms of a growing pattern?
You need the 100th term. Which kind of rule gets you there quickest?
What should you check before using any sequence rule?
Tap any card to flip it, or use Study as deck to go through them one at a time. In the full lesson these run as a spaced-repetition deck — you rate each card Hard, Good or Easy and the tricky ones keep coming back until they stick.
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