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Number patterns and linear sequences
Finding the rule and missing terms, and arithmetic versus geometric steps.
Start at 3 and keep adding 4: 3, 7, 11, 15, 19 … That list is a sequence. In this lesson you will learn how to:
- find the rule that makes a sequence
- continue a sequence, going up or going down
- find missing terms, even when there are several gaps in a row
- find a term far along a sequence, such as the 50th, without counting every step
- tell an arithmetic sequence from a geometric one
1. What a sequence is
Example: stepping stones
The 1st term is 3. To get to the next stone, you add 4. So the rule is “add 4”, and the next term is 19 + 4 = 23.
Going down
A sequence can go down too. Here the rule is take away 6.
Two names, one idea
A sequence that goes up or down by the same amount every time is called a linear sequence. It is also called an arithmetic sequence (say it a-rith-MET-ic). Both names mean the same thing.
Hop along
- Pick a number to start at and a rule. Then press Hop to make the frog jump to the next stone.
- Before each hop, say where the frog will land. Were you right?
2. Finding the rule
Example: find the next two terms of 5, 12, 19, 26
- Find every gap:12 − 5 = 7, 19 − 12 = 7, 26 − 19 = 7
- Every gap is 7, so the rule is:add 7
- Use the rule:26 + 7 = 33, 33 + 7 = 40
Going down: 61, 52, 43, 34
The terms get smaller, so the rule is a take away. 61 − 52 = 9, and the other gaps are 9 too. So the rule is take away 9, and the next term is 34 − 9 = 25.
Crack the rule
- Step 1: pick the rule. Step 2: use it to find the next two terms.
- Two with no slips moves you up a level. Level 1 adds, level 2 adds or takes away bigger numbers, and level 3 has decimal steps or goes below zero. The keypad gets a point key and a minus key on level 3.
3. Finding missing terms
A gap in the middle
The rule is add 6, so the missing term is 15 + 6 = 21. Check: 21 + 6 = 27.
A gap at the start
The rule is add 4. Going backwards, do the opposite: 9 − 4 = 5.
Example: two gaps in a row, 6, ?, ?, 21
- Count the jumps from 6 to 21:3 jumps
- How far is it from 6 to 21?21 − 6 = 15
- Share it between the jumps:15 ÷ 3 = 5, so the rule is add 5
Fill the gaps
- Fill each red ? stone. On the harder levels, count the jumps and share out the distance first.
- Two with no slips moves you up a level. Level 1 has a gap in the middle, level 2 a gap at the start, level 3 two gaps in a row, level 4 three gaps going down, and level 5 gives you just two terms.
4. Terms far along the sequence
Example: find the 20th term of 3, 7, 11, 15, 19
| Position | 1 | 2 | 3 | 4 | 5 | … | 20 |
|---|---|---|---|---|---|---|---|
| 4 × position | 4 | 8 | 12 | 16 | 20 | … | 80 |
| Change | − 1 | − 1 | − 1 | − 1 | − 1 | − 1 | |
| Sequence | 3 | 7 | 11 | 15 | 19 | … | 79 |
- The rule is add 4, so line it up with the 4 times table:4, 8, 12, 16, 20
- Each term is 1 less than the times table:term = 4 × position, then take away 1
- Use it for the 20th term:4 × 20 = 80, 80 − 1 = 79
- Check it with a term you know:the 5th term is 4 × 5 − 1 = 19
Which term is 59?
Undo the rule, backwards: add the 1 back, then divide by 4. So 59 + 1 = 60, and 60 ÷ 4 = 15. That makes 59 the 15th term.
Is 50 in the sequence?
50 + 1 = 51, and 51 is not in the 4 times table. So 50 is not in the sequence. (It falls between the 12th term, 47, and the 13th term, 51.)
Far away terms
- Find the step, line the sequence up with its times table, then find how far it is from the times table.
- Two with no slips moves you up a level. Level 1 finds terms up to the 20th, level 2 goes as far as the 100th, and level 3 works backwards: which term is it, and is it in the sequence at all?
5. Arithmetic or geometric?
Arithmetic: add 3
Geometric: multiply by 2
Geometric going down: divide by 2
Each term is half of the one before.
Neither: the step changes
The steps are 1, 2, 3, so the next step is 4: 7 + 4 = 11. When the step changes, look at the steps themselves. They often make a pattern of their own. In 1, 1, 2, 3, 5, 8, each term is the two before it added together, so the next term is 5 + 8 = 13.
Sort the sequences
- Look at every gap, then pick the kind of sequence. The hops show the steps once you have picked.
- Then find the next term.
6. Practice: pick your level
Start with Foundation if you are in Year 4 or just starting. Move up when you are getting most of them right.
The mistakes that cost marks
- Checking only the first gap. Saying the rule for 2, 4, 8, 16 is “add 2”. check every gap: 2, 4 and 8 are not the same. Each term is double the one before.
- Getting the sign wrong. Saying the rule for 40, 34, 28 is “add 6”. the terms get smaller, so the rule is take away 6.
- Using the rule the wrong way when going backwards. Filling ?, 9, 13, 17 with 13. going backwards, do the opposite: 9 − 4 = 5.
- Counting the missing terms instead of the jumps. Working out 15 ÷ 2 for 6, ?, ?, 21. there are 3 jumps from 6 to 21, so 15 ÷ 3 = 5.
- Mixing up the position and the term. Saying the 4th term of 5, 8, 11, 14 is 4. the 4th term is the 4th number in the list: 14.
- Forgetting the change in the times-table trick. Saying the 20th term of 3, 7, 11 is 4 × 20 = 80. each term is 1 less than the 4 times table: 80 − 1 = 79.
- Doubling to get a term twice as far along. Saying the 20th term is double the 10th term. use the rule for the 20th term itself: 4 × 20 − 1 = 79, not 2 × 39 = 78.
Quick checklist
Tick each one when you can do it without help.
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