Key ideas
Sequences
A sequence is a list of numbers that follows a rule. Each number is a term.
To continue one, find what happens each time: add, take away, double…
Describing in words
Say where it starts and what happens each time.
5, 8, 11, 14, … : start at 5 and add 3 each time.
The nth term
A formula like nth term = 4n + 1 gives any term straight away.
For the 10th term, put n = 10: 4 × 10 + 1 = 41.
Finding a formula
1. Make a table.
2. Find the difference: it goes up by 3, so start with 3 × S.
3. Compare and adjust: add or take away to make it fit.
4. Test it on a pattern you know.
Continuing a sequence
Write down the next two terms: 5, 9, 13, 17, …(2 marks)
What happens each time?5 → 9 → 13 → 17+4 each time✓1
In plain EnglishLook at the gap between each term and the next. Each time it goes up by 4.
Keep going.17 + 4 = 2121 + 4 = 25✓2
In plain EnglishAdd 4 to the last term, then add 4 again.
Write down the next two terms: 50, 44, 38, 32, …(2 marks)
What happens each time?−6 each time✓1
In plain EnglishEach term is 6 less than the one before.
Keep going.32 − 6 = 2626 − 6 = 20✓2
In plain EnglishTake 6 away from the last term, then take 6 away again.
Describing a sequence in words
Describe the sequence 3, 8, 13, 18, … in words.(1 mark)
Where does it start? What happens each time?First term: 3+5 each timeStart at 3 and add 5 each time.✓1
In plain EnglishA description in words needs two things: the first term, and what happens to get from one term to the next.
⚠ A common mistake: only saying "add 5". Lots of sequences add 5 (like 1, 6, 11, …), so you must say where it starts too.
Using a formula for the nth term
The nth term of a sequence is 4n + 1. Find (a) the 1st term (b) the 2nd term (c) the 10th term.(3 marks)
Put n = 1.4 × 1 + 1 = 5✓1
In plain Englishn stands for the position of the term. For the 1st term, n = 1.
Put n = 2.4 × 2 + 1 = 9✓2
In plain EnglishFor the 2nd term, n = 2. Multiply by 4 first, then add 1.
Put n = 10.4 × 10 + 1 = 41✓3
In plain EnglishYou don't need to write out all ten terms: the formula jumps straight there.
The nth term of a sequence is 3n + 2. Which term is equal to 62?(2 marks)
Make an equation.3n + 2 = 62✓1
In plain EnglishWe know the answer to the formula, so set it equal to 62.
Solve it.3n = 60n = 20, so it is the 20th term✓2
In plain EnglishTake 2 from both sides, then divide by 3.
Patterns and tables
Matchsticks are used to make squares in a row. Complete the table to show the number of matches for 4 and 5 squares.(2 marks)
Count the first few.
+3 each time✓1
| Squares (S) | 1 | 2 | 3 | 4 | 5 |
|---|---|---|---|---|---|
| Matches (M) | 4 | 7 | 10 | ? | ? |
In plain EnglishCount the matches in each pattern: 4, 7, 10. Each new square needs 3 more matches, because it shares a side with the one before.
Keep adding 3.10 + 3 = 1313 + 3 = 16✓2
In plain EnglishEach new square needs 3 more matches, so add 3 for 4 squares and 3 more for 5 squares.
Finding a formula
Find a formula for the number of matches, M, needed to make S squares.(2 marks)
Find the difference.
Goes up by 3, so try 3 × S✓1
| Squares (S) | 1 | 2 | 3 | 4 |
|---|---|---|---|---|
| Matches (M) | 4 | 7 | 10 | 13 |
| 3 × S | 3 | 6 | 9 | 12 |
In plain EnglishThe matches go up by 3 each time, so the formula starts with 3 times S. Write 3 × S under each column to compare.
Compare and adjust.3 × S is always 1 less than MM = 3S + 1✓2
In plain EnglishEach number of matches is 1 more than 3 × S, so add 1.
Test it.S = 2: 3 × 2 + 1 = 7 ✓
In plain EnglishAlways check your formula on a pattern you know. 2 squares needs 7 matches, which matches the table.
⚠ A common mistake: writing M = S + 3. It goes up by 3, so you multiply S by 3.
Using your formula
Use M = 3S + 1 to find (a) how many matches are needed for 20 squares (b) how many squares can be made with 100 matches.(3 marks)
Put S = 20.M = 3 × 20 + 1 = 61 matches✓1
In plain EnglishReplace S with 20 in the formula.
Backwards: make an equation.3S + 1 = 1003S = 99✓2S = 33 squares✓3
In plain EnglishNow we know M = 100. Take 1 away from both sides, then divide by 3.
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Your turn
Type numbers for terms. For a formula, type it with the letters, like M = 3S + 1, and check the preview under the box. Stuck? Tap Need a hint?