Numbers and sequences
Sequences
Find the jump between one term and the next, and you can carry a sequence on forever — forwards, backwards, and straight through zero.
Terms and the jump size
A SEQUENCE is a list of numbers following a rule. Each number is a TERM. In a LINEAR sequence the jump between terms is always the same size. Find that jump and you can continue the sequence as far as you like. 3, 7, 11, 15, … the jump is +4. 20, 17, 14, 11, … the jump is −3.
Finding the jump size
Subtract any term from the one after it. For 3, 7, 11: 7 − 3 = 4, and 11 − 7 = 4. The jump is +4. Always check with a second pair. If the two gaps do not match, the jump is not constant and it is not a linear sequence.
Counting back through zero
A sequence going down does not stop at zero — it carries on into NEGATIVE numbers. 8, 5, 2, −1, −4, … The jump is still −3 every time. After 2 comes 2 − 3 = −1, then −1 − 3 = −4. The rule does not change just because you have crossed zero.
Three things define a sequence
To write a linear sequence you need exactly three pieces of information: 1. the FIRST TERM 2. the JUMP SIZE 3. how many TERMS you want First term 1, jump +4, 5 terms → 1, 5, 9, 13, 17. First term −15, jump +11, 5 terms → −15, −4, 7, 18, 29. With all three you can write the sequence out exactly; with only two you cannot.
Worked examples
Finding the jump size
- Take the sequence 21, 28, 35, 42.
- Subtract: 28 − 21 = 7.
- Check another pair: 35 − 28 = 7. The jump is constant.
Answer: The jump size is +7
Continuing a sequence backwards
- The sequence is 20, 17, 14, 11.
- The jump is −3.
- Keep going: 11 − 3 = 8, 8 − 3 = 5.
Answer: The next two terms are 8 and 5
Counting back through zero
- Start at 8 with a jump of −3.
- 8, 5, 2 — the next is 2 − 3.
- That takes you below zero.
Answer: 8, 5, 2, −1, −4
Writing a sequence from its three facts
- First term −15, jump +11, 5 terms.
- Start at −15, then add 11 each time.
- −15 + 11 = −4, −4 + 11 = 7, and so on.
Answer: −15, −4, 7, 18, 29
Finding a missing term
- The sequence is 6, ▢, 16, 21.
- From 16 to 21 the jump is +5.
- So the missing term is 6 + 5.
Answer: The missing term is 11
Where you see this in real life
- Saving the same amount each week gives a sequence with a constant jump.
- Temperatures falling by the same amount each hour count back through zero into negative numbers.
- Seat rows numbered in equal steps let you find any row without counting them all.
Key words
- Sequence
- A list of numbers that follows a rule.
- Term
- One number in a sequence.
- Jump size
- The amount added or subtracted between terms.
- Linear sequence
- A sequence where the jump size is always the same.
- Negative number
- A number less than zero.
- First term
- The number a sequence starts with.