C2.7 – Sequences

Section A — Recall    Questions 1–5
1. State what is meant by a sequence.
Answer
A sequence is a list of numbers written in a specific order, following a rule. Each number in the list is called a term.
2. State what the term-to-term rule of a sequence tells you.
Answer
It tells you how to get from one term to the next term in the sequence.
3. State the term-to-term rule for the sequence: 6, 10, 14, 18, …
Answer
Add 4.
4. State the general formula for the nth term of a linear sequence, in terms of $d$ and $c$.
Answer
$n\text{th term} = dn + c$
Remember: $d$ is the common difference and $c$ is the first term minus $d$.
5. A quadratic sequence has a constant second difference. State how you use this second difference to find the coefficient of $n^2$ in the nth term.
Answer
Divide the second difference by 2.
Section B — Application    Questions 6–10
6. Write the next two terms of the sequence: 4, 9, 14, 19, …
Answer
  • Term-to-term rule: add 5
  • $19 + 5 = 24$,   $24 + 5 = 29$
24, 29
7. Find the nth term of the linear sequence: 4, 7, 10, 13, …
Answer
  • Common difference: $d = 7 – 4 = 3$, so the nth term is based on $3n$
  • Write the 3 times table above the sequence and see how to get from one to the other
3
6
9
12
↓+1
↓+1
↓+1
↓+1
4
7
10
13
$3n + 1$
$3n + 1$
Check: when $n = 1$: $3(1) + 1 = 4$ ✓   when $n = 4$: $3(4) + 1 = 13$ ✓
8. The nth term of a sequence is $2n + 3$. Find the 15th term.
Answer
  • Substitute $n = 15$
  • $2(15) + 3 = 30 + 3 = 33$
33
9. The nth term of a sequence is $n^3 – 2$. Find the 4th term.
Answer
  • Substitute $n = 4$
  • $4^3 – 2 = 64 – 2 = 62$
62
10. Find the nth term of the quadratic sequence: 3, 6, 11, 18, …
Answer
  • Find the first and second differences
3 6 11 18 3 5 7 2 2

First difference: $6-3=3$, $11-6=5$, $18-11=7$ — not constant.
Second difference (highlighted): $5-3=2$, $7-5=2$ — always 2, so it’s a quadratic sequence.

  • Coefficient of $n^2$: $a = 2 \div 2 = 1$, so the sequence starts with $n^2$
  • Write out $n^2$ underneath the sequence, and subtract
Sequence minus $n^2$
3
6
11
18
−1
−4
−9
−16
2
2
2
2

The new sequence is constant: 2, 2, 2, 2, so nth term $= n^2 + 2$

$n^2 + 2$
Check: when $n = 4$: $4^2 + 2 = 18$ ✓
Section C — Challenge    Questions 11–15
11. For each sequence below, write the next two terms and state the term-to-term rule.
(a)
40, 34, 28, 22, …
(b)
3, 6, 12, 24, …
(c)
2, 3, 5, 8, 12, …
Answer
(a)

Rule: subtract 6.   $22 – 6 = 16$,   $16 – 6 = 10$

16, 10
(b)

Rule: multiply by 2.   $24 \times 2 = 48$,   $48 \times 2 = 96$

48, 96
(c)

Rule: add one more each time (add 1, then 2, then 3, then 4, …). The differences are 1, 2, 3, 4, so the next differences are 5 and 6.   $12 + 5 = 17$,   $17 + 6 = 23$

17, 23
12. Consider the linear sequence: 12, 9, 6, 3, …
(a)
Find the nth term of this sequence.
(b)
Use your nth term to find the 10th term.
(c)
Is $-100$ a term in this sequence? Show working to support your answer.
Answer
(a) nth term
  • Common difference: $d = 9 – 12 = -3$. Because $d$ is negative, write out the negative 3 times table.
−3
−6
−9
−12
↓+15
↓+15
↓+15
↓+15
12
9
6
3
$-3n + 15$
−3 (+3)→ 0 (+12)→ 12
$-3n + 15$
(b) 10th term

$-3(10) + 15 = -30 + 15 = -15$

$-15$
(c) Is $-100$ a term?
  • Set $-3n + 15 = -100$
  • $-3n = -115$, so $3n = 115$
  • Check nearby multiples of 3:   $3 \times 38 = 114$   and   $3 \times 39 = 117$
  • 115 is not one of these — it falls between $3 \times 38$ and $3 \times 39$
No. There is no whole number $n$ with $3n = 115$, so $-100$ is not a term in this sequence.
13. Consider the quadratic sequence: 5, 8, 13, 20, 29, …
(a)
Find the nth term of this sequence.
(b)
Use your nth term to find the 8th term.
Answer
(a) nth term
  • Find the first and second differences
5 8 13 20 29 3 5 7 9 2 2 2

First difference: $8-5=3$, $13-8=5$, $20-13=7$, $29-20=9$ — not constant.
Second difference (highlighted): $5-3=2$, $7-5=2$, $9-7=2$ — always 2, so it’s a quadratic sequence.

  • Coefficient of $n^2$: $a = 2 \div 2 = 1$, so the sequence starts with $n^2$
  • Write out $n^2$ underneath the sequence, and subtract
Sequence minus $n^2$
5
8
13
20
29
−1
−4
−9
−16
−25
4
4
4
4
4

The new sequence is constant: 4, 4, 4, 4, 4, so nth term $= n^2 + 4$

$n^2 + 4$
(b) 8th term

$8^2 + 4 = 64 + 4 = 68$

68
14. Consider the cubic sequence: 0, 7, 26, 63, …
(a)
Find the nth term of this sequence.
(b)
Use your nth term to find the 6th term.
(c)
Is 999 a term in this sequence? Show working to support your answer.
Answer
(a) nth term
  • Find the first, second, and third differences
0 7 26 63 7 19 37 12 18 6

First difference: $7-0=7$, $26-7=19$, $63-26=37$ — not constant.
Second difference: $19-7=12$, $37-19=18$ — still not constant.
Third difference (blue): $18-12=6$ — this only gives one value here (just 4 terms), but it tells us the sequence is cubic.

  • Coefficient of $n^3$: $a = 6 \div 6 = 1$, so the sequence starts with $n^3$
  • Write out $n^3$ underneath the sequence, and subtract
Sequence minus $n^3$
0
7
26
63
−1
−8
−27
−64
−1
−1
−1
−1

The new sequence is constant: $-1, -1, -1, -1$, so nth term $= n^3 – 1$

$n^3 – 1$
(b) 6th term

$6^3 – 1 = 216 – 1 = 215$

215
(c) Is 999 a term?
  • Set $n^3 – 1 = 999$
  • $n^3 = 1000$
  • $n = 10$, since $10^3 = 1000$
Yes. $n = 10$ is a whole number, so 999 is the 10th term.
15. Sequence A has nth term $n^2$. Sequence B is formed by adding 2 to every term of Sequence A.
(a)
Write the first four terms of Sequence A, then the first four terms of Sequence B.
(b)
Write the nth term of Sequence B.
(c)
Explain how the relationship between Sequence A and Sequence B helped you find the nth term of Sequence B.
Answer
(a) First four terms

Sequence A ($n^2$): $1^2=1$, $2^2=4$, $3^2=9$, $4^2=16$ → 1, 4, 9, 16

Sequence B (add 2 to each term of A): 3, 6, 11, 18

A: 1, 4, 9, 16    B: 3, 6, 11, 18
(b) nth term of B
$n^2 + 2$
(c) Explanation
Every term of Sequence B is 2 more than the matching term of Sequence A. Since Sequence A’s nth term is $n^2$, adding 2 to it gives Sequence B’s nth term: $n^2 + 2$.
Key idea: if you know how one sequence relates to another (e.g. “add 2 to every term”), you can apply the same change to its nth term to find the nth term of the related sequence.
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