Revision mat · Sequences & Patterns

No calculator15 marks

Question mix weighted by 95 real exam questions from 64 GCSE papers (2017–24), so the most common question types get the most space. Show your working.

Term-to-term rule $\text{+ or }-\text{ the same each time}$Common difference $\text{next term}-\text{this term}$Square numbers $1,\ 4,\ 9,\ 16,\ 25,\ \ldots$Triangular numbers $1,\ 3,\ 6,\ 10,\ 15,\ \ldots$Fibonacci-type $\text{add the previous two terms}$
Continue & rule≈44% of real exam Qs
1In this number chain the same two steps repeat: “$\times 3$ then $-3$”, then “$+3$”, alternating.
The chain so far is:
$6, \; 15, \; 18, \; 51, \; 54, \; \boxed{?}$
The next step is “$\times 3$ then $-3$”.
Work out the next number in the chain.
Answer: (3 marks)
2The population of a town is modelled by $P_{n+1} = a\,P_n + 800$, where $P_n$ is the population at the start of year $n$.
At the start of year 1 the population is $20000$, and at the start of year 2 it is $24800$.
Use the model to work out the population at the start of year 4.
Answer: (3 marks)
Fibonacci-type≈13% of real exam Qs
3Each term of this Fibonacci-type sequence is the sum of the two terms before it.
$3,\ 6,\ 9,\ \boxed{\phantom{0}},\ 24$
Work out the missing term.
Answer: (2 marks)
Missing terms≈11% of real exam Qs
4Here is a sequence with one term missing.
$54,\ \boxed{\phantom{00}},\ 46,\ 42,\ 38$
Work out the value of the missing term.
Answer: (2 marks)
Special sequences≈11% of real exam Qs
5The $n$th term of a sequence is $n^2 + 4$.
Work out the smallest integer $n$ for which the $n$th term is greater than $500$.
Answer: (2 marks)
★ Exam capstonemixed & other forms ≈22% of real exam Qs
6In this sequence, each term after the second is the sum of the two terms before it.
The 1st, 3rd, 4th and 5th terms are
$3, \; \boxed{?}, \; 7, \; 11, \; 18$
Work out the missing 2nd term.
Answer: (3 marks)

Mark scheme · Sequences & Patterns

Total: 15 marks

Award the marks shown for each correct step, then add up the total out of 15. A method mark counts even if the final answer is wrong.

1In this number chain the same two steps repeat: “$\times 3$ then $-3$”, then “$+3$”, alternating.
The chain so far is:
$6, \; 15, \; 18, \; 51, \; 54, \; \boxed{?}$
The next step is “$\times 3$ then $-3$”.
Work out the next number in the chain.[3 marks]
Method
The steps alternate: “$\times 3$ then $-3$”, then “$+3$”.
The next step is “$\times 3$ then $-3$”: $54 \times 3 - 3 = 159$.
Answer: $159$
Marks
✔1 markIdentifies the next step is ×3 then −3
✔1 mark54 × 3 − 3
✔1 mark159
2The population of a town is modelled by $P_{n+1} = a\,P_n + 800$, where $P_n$ is the population at the start of year $n$.
At the start of year 1 the population is $20000$, and at the start of year 2 it is $24800$.
Use the model to work out the population at the start of year 4.[3 marks]
Method
Find $a$: $24800 = a\times20000 + 800$, so $a = (24800 - 800) \div 20000 = 1.2$.
Year 3: $P_3 = 1.2\times24800 + 800 = 30560$.
Year 4: $P_4 = 1.2\times30560 + 800 = 37472$.
Answer: $37472$
Marks
✔1 marka = 1.2
✔1 markyear 3 population 30560
✔1 markyear 4 population 37472
3Each term of this Fibonacci-type sequence is the sum of the two terms before it.
$3,\ 6,\ 9,\ \boxed{\phantom{0}},\ 24$
Work out the missing term.[2 marks]
Method
Each term is the sum of the two before it, so a term equals the next term minus the term before it.
$24 - 9 = 15$.
Answer: $15$
Marks
✔1 markUse term = next − previous (or build forward)
✔1 markMissing term 15
4Here is a sequence with one term missing.
$54,\ \boxed{\phantom{00}},\ 46,\ 42,\ 38$
Work out the value of the missing term.[2 marks]
Method
The term-to-term rule is: subtract 4.
The missing term is $50$.
Answer: $50$
Marks
✔1 markTerm-to-term rule: subtract 4
✔1 markMissing term 50
5The $n$th term of a sequence is $n^2 + 4$.
Work out the smallest integer $n$ for which the $n$th term is greater than $500$.[2 marks]
Method
Solve $n^2 + 4 > 500$, i.e. $n^2 > 496$.
$\sqrt{496} \approx 22.3$, so the smallest whole number is $n = 23$.
Check: $23^2 + 4 = 533 > 500$, but $22^2 + 4 = 488 \le 500$.
Answer: $23$
Marks
✔1 markn² > 496
✔1 markn = 23
6In this sequence, each term after the second is the sum of the two terms before it.
The 1st, 3rd, 4th and 5th terms are
$3, \; \boxed{?}, \; 7, \; 11, \; 18$
Work out the missing 2nd term.[3 marks]
Method
The 3rd term $=$ 1st $+$ 2nd, so $7 = 3 + (\text{2nd term})$.
2nd term $= 7 - 3 = 4$.
Check: $4 + 7 = 11$ ✓.
Answer: $4$
Marks
✔1 mark3rd = 1st + 2nd
✔1 mark2nd = 7 − 3
✔1 mark4
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