Sequences flashcards
11 cards on Sequences: the key points, worked examples and the slips to watch. Swipe through them, or print them.
Card 1 of 11
Sequences
Term-to-term rules
How do you find the term-to-term rule of a sequence and continue it?
Answer it in your head, then turn the card over.
Term-to-term rules
Key points
- The term-to-term rule says how to get from one term to the next, such as "add 3".
- Work out the gaps between neighbouring terms first.
- Check the rule works for every gap, not just the first one.
- Some rules have two steps, such as "double it, then add 1".
2, 5, 11, 23, …
Each term is double the one before, plus 1
Next term: 23 × 2 + 1 = 47
Watch out
In 1, 2, 4, 8 the first gap suggests "add 1", but only "double it" fits every gap.
Sequences
Arithmetic and geometric sequences
What is the difference between an arithmetic sequence and a geometric sequence?
Answer it in your head, then turn the card over.
Arithmetic and geometric sequences
Key points
- Arithmetic (linear): add or subtract the same number each time, the common difference.
- Geometric: multiply by the same number each time.
- A geometric sequence can multiply by a fraction: 80, 40, 20, 10 multiplies by .
Example
7, 11, 15, 19 adds 4: arithmetic, next 23
2, 6, 18, 54 multiplies by 3: geometric, next 162
Watch out
2, 6, 18, 54 is not arithmetic: its gaps (4, 12, 36) are not all the same.
Sequences
Square, cube and triangular numbers
What are the square, cube and triangular number sequences?
Answer it in your head, then turn the card over.
Square, cube and triangular numbers
Key points
- Square numbers: 1, 4, 9, 16, 25, from n × n.
- Cube numbers: 1, 8, 27, 64, 125, from n × n × n.
- Triangular numbers: 1, 3, 6, 10, 15, 21, the dots in growing triangles.
- The triangular gaps grow by one each time: +2, +3, +4, +5.
Example
The 6th triangular number: 15 + 6 = 21
The 5th cube number: 5 × 5 × 5 = 125
Watch out
Square numbers are n × n, not n × 2: the 3rd square number is 9, not 6.
Sequences
Fibonacci-type sequences
How is each term of a Fibonacci-type sequence found?
Answer it in your head, then turn the card over.
Fibonacci-type sequences
Key points
- Each term is the two terms before it added together.
- The Fibonacci sequence itself is 1, 1, 2, 3, 5, 8, 13.
- Any two starting numbers make a Fibonacci-type sequence.
3, 4, 7, 11, …
Next: 7 + 11 = 18
Then: 11 + 18 = 29
Watch out
Adding the last gap again (11 + 4 = 15) treats it as arithmetic: add the two terms before.
Sequences
Finding the nth term
How do you find the nth term of an arithmetic sequence?
Answer it in your head, then turn the card over.
Finding the nth term
Key points
- Find the common difference: it is the number in front of n.
- Write out that times table, then see what to add or take away to reach each term.
- Check the rule with the first two terms: put in n = 1 and n = 2.
7, 11, 15, 19
The difference is 4, so start with 4n: 4, 8, 12, 16
Each term is 3 more, so the nth term is 4n + 3
Check: n = 1 gives 4 + 3 = 7
Watch out
The difference goes in front of n, not on the end: 7, 11, 15, 19 is 4n + 3, not n + 4.
Sequences
The nth term of a falling sequence
How do you find the nth term of a sequence that goes down?
Answer it in your head, then turn the card over.
The nth term of a falling sequence
Key points
- A sequence that goes down has a negative difference, so start with −3n, −5n and so on.
- Compare with the terms of that negative times table, then add what is needed.
- It is usually neatest written with the number first: 23 − 3n.
20, 17, 14, 11
Difference −3, so start with −3n: −3, −6, −9, −12
Each term is 23 more, so the nth term is 23 − 3n
Check: n = 4 gives 23 − 12 = 11
Watch out
A falling sequence never has +3n: 3n + 17 would give 20, 23, 26, going up.
Sequences
Sequences from patterns
How do you find the nth term of a pattern of shapes, such as matchsticks?
Answer it in your head, then turn the card over.
Sequences from patterns
Key points
- Count the matchsticks (or tiles) in the first few patterns to make a sequence.
- Find the nth term of that sequence in the usual way.
- The difference is what each new pattern adds: the number in front of n.
Patterns of 4, 7, 10 matchsticks
Each pattern adds 3, so start with 3n: 3, 6, 9
One more each time: the nth term is 3n + 1
Pattern 20: 3 × 20 + 1 = 61 matchsticks
Watch out
Do not draw pattern 20: use the nth term, which jumps straight there.
Sequences
Using the nth term
How do you use an nth term to find any term of a sequence?
Answer it in your head, then turn the card over.
Using the nth term
Key points
- Substitute the position for n: n = 1 for the first term, n = 50 for the 50th.
- The nth term finds any term directly, with no need to list the ones before it.
- Multiply before you add or subtract.
Example
50th term of 4n + 3: 4 × 50 + 3 = 203
100th term of 3n − 2: 3 × 100 − 2 = 298
Watch out
When a question asks for the nth term, "add 4" will not do: give the rule in n, such as 4n + 3.
Sequences
Testing whether a number is a term
How do you decide whether a number is a term of a sequence?
Answer it in your head, then turn the card over.
Testing whether a number is a term
Key points
- Set the nth term equal to the number and solve for n.
- A whole-number n means yes, and n tells you which term it is.
- Any other n means no: the sequence jumps over that number.
Example
5n + 2 = 87: 5n = 85, so n = 17. Yes, the 17th term
4n + 3 = 150: 4n = 147, so n = 36.75. No
Watch out
Never round n: 36.75 does not make 150 the 37th term, which is 151.
Sequences
HigherSpotting a quadratic sequence
How can you tell that a sequence is quadratic, and what goes in front of n²?
Answer it in your head, then turn the card over.
Spotting a quadratic sequence
Key points
- Work out the first differences, then the differences between those.
- If the second differences are all the same, the sequence is quadratic.
- The number in front of n² is half the second difference.
4, 13, 28, 49
First differences: 9, 15, 21
Second differences: 6, 6
6 ÷ 2 = 3, so the nth term starts 3n²
Watch out
Halve the second difference: a second difference of 6 gives 3n², not 6n².
Sequences
HigherThe nth term of a quadratic sequence
How do you find the full nth term of a quadratic sequence?
Answer it in your head, then turn the card over.
The nth term of a quadratic sequence
Key points
- Find the n² term: half the second difference.
- Take that n² term away from each term of the sequence.
- Find the nth term of what is left (it is linear), then add it on.
- Check by putting n = 1 and n = 2 into your answer.
3, 8, 15, 24
Second difference 2, so start with n²: 1, 4, 9, 16
3 − 1, 8 − 4, 15 − 9, 24 − 16 gives 2, 4, 6, 8 = 2n
nth term: n² + 2n. Check n = 3: 9 + 6 = 15
Watch out
After taking away n², what is left can still change: find its nth term, not just one number.
Print these cards
Four cards to an A4 sheet, each the size of a standard A6 revision card.
One-sided
Each card prints on one face: the question at the top, the notes under it.
Two-sided
Questions on the front, notes on the back. Print double-sided and flip on the long edge, then cut along the lines.
Put it into practice
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