KS3 · Maths

Generating sequences from position-to-term rules

What's the 1,000th term of a sequence? You can't count on that far. A rule that starts from the term number gets you there in one step.

Sort the rules

Two questions to ask about any rule

Pick a card, then pick where it belongs. The sequence we are trying to make is 9, 11, 13.

Does this rule start from the term number, or from the previous term?

Still to sort

Position-to-term (0)

Starts from the term number.

Where the line is: If you could work out T50 without knowing T49, it starts from the term number.

Term-to-term (0)

Starts from the previous term.

Where the line is: If you need the term before to get the next one, it is term-to-term.

5 of 5 still to sort.

Every card is a rule. Sort them twice: first by where the rule starts, then by whether it really makes 9, 11, 13.

Commit first

Pattern 1 is one triangle, pattern 2 is two triangles, and so on. The number of lines goes 3, 6, 9, 12.

How many lines are in the 100th pattern?

Generating terms from a rule

Problem

Use the rule 'multiply the term number by 5, then subtract 2' to find T1 to T5 and T10. Then try the rule 'cube the term number, then add 1'.

Which rule do you believe?

Building the rule for 6, 11, 16, 21, 26

The sequence is 6, 11, 16, 21, 26. It goes up by 5 each time.

Which of these is the position-to-term rule for this sequence?
How sure are you?

Your turn to fill the gaps

Find the rule, then use it

Find the position-to-term rule for 12, 21, 30, 39, then use it to find T50.

  1. Here is the method on a different sequence, 6, 10, 14, 18. It goes up by 4. T1 = 2 + 4 and T2 = 2 + 4 + 4, so T10 is 2 add 10 lots of 4. The rule is multiply the term number by 4, then add 2.
  2. Now yours: 12, 21, 30, 39. It goes up by 9 each time. First, write T1 as a starting value plus one difference.
  3. missing step
Which line is step 3?

Hunt the slip

Four rules, one slip

A student worked out the 60th term for four different rules. Three are right. Which line has the mistake?

A student's answers — which line goes wrong?

WHAT YOU'VE LEARNED

A quick recap of today's lesson.

Start from the term number, not the term before it.

What you need to know

  • Terms are named by position: T10 is the 10th term. A position-to-term rule starts from the term number. Put the position in, and the term comes out.
  • A term-to-term rule starts from the previous term instead. To reach T100 you would have to go through every term before it.
  • Because a position-to-term rule starts from the term number, it jumps straight to T10, T50 or T100.
  • Have a goZak is asked for T50 of 'multiply the term number by 7'. He starts writing out 7, 14, 21... on a long sheet of paper. How many terms does he actually need to write down?

    None. T50 = 7 × 50 = 350.

    The rule starts from the term number, so 50 goes straight in. Zak has built a term-to-term method by mistake.

  • Follow the steps in the order the rule gives them. 'Multiply by 5, then subtract 2' means multiply first.
  • If a sequence goes up by the same amount each time, write each term as a starting value plus lots of that difference.
  • T1 is the starting value plus one difference, T2 is the starting value plus two, and so on. The term number says how many differences.
  • Have a goA sequence has a starting value of 5 and a difference of 2. How many 2s are added to 5 to make T8, and what is T8?

    Eight 2s, so T8 = 5 + 16 = 21.

    The term number says how many differences are added, so T8 has eight of them.

  • So the rule is: multiply the term number by the difference, then add the starting value. The starting value is T1 take away one difference.
  • Have a goA sequence has T1 = 11 and goes up by 4. What do you add after multiplying the term number by 4?

    Add 7.

    The starting value is T1 take away one difference: 11 − 4 = 7. Adding 11 would count the first difference twice.

  • A rule is only right if it works for every term number you test. Check them all.
  • Two sequences can go up by the same amount and still have different rules, because their starting values differ.

The big picture

A position-to-term rule starts from the term number, so it can jump straight to T10, T50 or T100. When a sequence goes up by the same amount each time, you can find the rule by writing each term as a starting value plus lots of that difference. A rule is only right if it works for every term number you test.

Key points

1A position-to-term rule starts from the term number. A term-to-term rule starts from the previous term.
2A position-to-term rule jumps straight to any term, such as T10, T50 or T100.
3For a sequence with a constant difference, write each term as a starting value plus lots of the difference.
4The rule is: multiply the term number by the difference, then add the starting value (T1 take away one difference).
5A rule is only right if it works for every term number you test.

Worked example

Problem

A student says the rule for 5, 7, 9, 12 is 'multiply the term number by 2, then add 3'. Test it on every term. Does it work?

⚠ Watch out

Checking a rule against T1 only. A rule can get the first term right and still be wrong: 'multiply the term number by 9' gives 9 for T1 when the sequence is 9, 11, 13, but 18 for T2.

🧠

Memory hook

Term number in, term out. Multiply by the difference, add the start, and check every term.

✓

Check yourself

Rule: multiply the term number by 6, then subtract 4. Find T3 and T20. Why didn't you need T19? (T3 = 14, T20 = 116. The term number goes straight in.)

Flashcards

(12)
What is a position-to-term rule?
A rule that starts from the term number. Put the position in, and the term comes out.
What is a term-to-term rule?
A rule that starts from the previous term. You need the term before to get the next one.
Why can a position-to-term rule jump to T100?
Because it starts from the term number, so 100 goes straight in. No earlier terms are needed.
What does T10 mean?
The 10th term of the sequence.
What is T1 for 'multiply the term number by 4, then subtract 3'?
T1 = 4 × 1 − 3 = 1. Put the term number in and follow the steps in the order given.
A sequence goes up by 5 and has a starting value of 2. Write T2 in full.
T2 = 2 + 5 + 5 = 12. The term number says how many differences are added.
How do you build the rule for a sequence with a constant difference?
Multiply the term number by the difference, then add the starting value.
How do you find the starting value?
Take one difference away from T1.
Why is 'first term add the difference' wrong?
The first term already contains one difference, so adding the first term counts the difference twice.
How many terms should you check a rule against?
Every term you have. A rule can match T1 and still fail from T2 onwards.
Two sequences both go up in 3s. Must they have the same rule?
No. They multiply by 3 either way, but they add different starting values if they start in different places.
What does 'cube the term number' mean?
Multiply the term number by itself, and then by itself again. For term number 4, that is 4 × 4 × 4 = 64.

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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