These problems use term-to-term rules, missing terms, special number sequences and Fibonacci-type sequences. A worked example shows how to start; hints are at the foot.
Worked example: how to start
A sequence starts $4, 7, 10, 13, \ldots$ Work out the $8$th term.
1A sequence starts $6, 10, 14, 18, \ldots$ Work out the $7$th term.[2 marks]
Answer:
2In the sequence $5,\ \square,\ \square,\ 20,\ \ldots$ the terms increase by the same amount each time. Work out the two missing terms.[3 marks]
Answer:
3The first two terms of a Fibonacci-type sequence are $3$ and $4$. Work out the $5$th term.[3 marks]
Answer:
4Which is the first square number that is greater than $50$?[2 marks]
Answer:
5The $10$th triangular number is $55$ and the $11$th is $66$. Work out the $12$th triangular number.[2 marks]
Answer:
★Extension: A Fibonacci-type sequence is $x,\ 5,\ x+5,\ \ldots$ If the $4$th term is $13$, work out $x$.
Stuck? Hints (don't peek unless you need to)1. Find the common difference, then keep adding it.2. From $5$ to $20$ is $15$ across $3$ steps, so each step is $15\div3$.3. Each term is the sum of the previous two.4. Square numbers are $1, 4, 9, 16, 25, 36, 49, 64, \ldots$5. The gaps between triangular numbers go up by $1$: $\ldots, +11, +12, \ldots$
Solutions & mark scheme · Sequences & Patterns
Total: 12 marks
Award the marks shown for each correct step; many of these have more than one valid route, so give method marks for any correct working.
1A sequence starts $6, 10, 14, 18, \ldots$ Work out the $7$th term.[2]
Model solution
Rule: add $4$.
$22$ (5th), $26$ (6th), $30$ (7th).
Answer: $30$
Marks
✔1Common difference $=4$
✔1$7$th term $=30$
2In the sequence $5,\ \square,\ \square,\ 20,\ \ldots$ the terms increase by the same amount each time. Work out the two missing terms.[3]
Model solution
From $5$ to $20$ is $15$, over $3$ steps: each step $=15\div3=5$.
Missing terms: $5+5=10$ and $10+5=15$.
Answer: $10\text{ and }15$
Marks
✔1Step $=5$
✔1First missing $=10$
✔1Second missing $=15$
3The first two terms of a Fibonacci-type sequence are $3$ and $4$. Work out the $5$th term.[3]
Model solution
$3,\ 4,\ 7$ (3rd), $11$ (4th), $18$ (5th).
Answer: $18$
Marks
✔1$3$rd term $=7$
✔1$4$th term $=11$
✔1$5$th term $=18$
4Which is the first square number that is greater than $50$?[2]
Model solution
Square numbers: $\ldots, 49, 64, \ldots$
The first one over $50$ is $64$ ($=8^2$).
Answer: $64$
Marks
✔1Lists square numbers past $50$
✔1$=64$
5The $10$th triangular number is $55$ and the $11$th is $66$. Work out the $12$th triangular number.[2]
Model solution
From the $11$th to the $12$th you add $12$.
$66+12=78$.
Answer: $78$
Marks
✔1Adds $12$
✔1$=78$
★Extension
The $3$rd term is $x+5$; the $4$th term is $5+(x+5)=x+10$.
So $x+10=13$, giving $x=3$.
Check: $3, 5, 8, 13$ – a Fibonacci-type sequence. ✓
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