Sequences

1. Sequences
Dr J Frost (jfrost@tiffin.kingston.sch.uk)
www.drfrostmaths.com
Objectives:
1. Understand term-to-term vs position-to-term rules.
2. Be able to generate terms of a sequence given a formula.
Find the formula for a linear sequence.

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3. Guidance
Suggested Lesson Structure:
Lesson 1: Generating sequences (term-to-term, position-to-
term)
Lesson 2: Finding 𝑛th term formula for linear sequences
Lesson 3: Pictorial Sequence Activity
Lesson 4: End-of-topic Assessment
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4. STARTER :: What’s next in each sequence?
6, 13, 20, 27, 𝟑𝟒, 𝟒𝟏, …
4, 2
1
2
, 1, −
𝟏
𝟐
, −𝟐, …
4, 12, 36, 𝟏𝟎𝟖, 𝟑𝟐𝟒, …
4, 6, 9, 13, 𝟏𝟖, 𝟐𝟒, …
2, 5, 7, 12, 19, 𝟑𝟏, 𝟓𝟎, …
5, 25, 15, 75, 65, 𝟑𝟐𝟓, 𝟑𝟏𝟓, …
1, 8, 27, 64, 𝟏𝟐𝟓, 𝟐𝟏𝟔, …
243, 27, 9, 3, 3, 𝟏, …
A sequence is simply an ordered list of items (possibly infinitely long),
usually with some kind of pattern. What are the next two terms in each
sequence?
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Only 1 term needed.
(Nicked off 2015’s
‘Child Genius’ on
Channel 4)
Divide one term by the
next to get the one after
that.
a
b
c
d
e
f
g
h

5. Term-to-term rules
Some sequences we can generated by stating a rule to say
how to generate the next term given the previous term(s).
Description First 5 terms
The first term of a sequence is 1.
+3 to each term to get the next.
1, 4, 7, 10, 13
The first term of a sequence is 3.
× 2 to each term to get the next.
3, 6, 12, 24, 48
The first two terms are 0 and 1.
Add the last two terms to get the next.
0, 1, 1, 2, 3
(known as the Fibonacci
sequence)
?
?
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What might be the disadvantage of using a term-to-term rule?
To get a particular term in the sequence, we have to
generate all the terms in the sequence before it. This is
rather slow if you say want to know the 1000th term!
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6. [JMC 2009 Q11] In a sequence of numbers, each term after
the first three terms is the sum of the previous three terms.
The first three terms are -3, 0, 2. Which is the first term to
exceed 100?
A 11th term B 12th term C 13th term
D 14th term E 15th term
JMC Puzzle
C
B
A D E
Terms are: -3, 0, 2, -1, 1, 2, 2, 5, 9, 16, 30, 55, 101

7. Position-to-term :: ‘𝒏th term’
It’s sometimes more helpful to be able to generate a term of a formula based on
its position in the sequence.
We could use it to say find the 300th term of a sequence without having to write all
the terms out!
We use 𝑛 to mean the position in the sequence. So if we want the 3rd term,
𝑛 = 3.
𝒏th term 1st term 2nd term 3rd term 4th term
𝟑𝐧 3 6 9 12
𝟓𝐧 5 10 15 20
𝟐𝐧 − 𝟏 1 3 5 7
𝐧𝟐
+ 𝟏 2 5 10 17
𝐧 𝐧 + 𝟏
𝟐
1 3 6 10
𝟐𝒏
2 4 8 16
This formula
gives the
triangular
numbers!
So 3𝑛 gives the
3 times table, 5𝑛
the five times
table, and so on.
? ? ? ?
? ? ? ?
? ? ? ?
? ? ? ?
? ? ? ?
? ? ? ?

8. Check Your Understanding
Find the first 4 terms in each of these sequences, given the
formula for the 𝑛th term.
4𝑛 + 3 → 𝟕, 𝟏𝟏, 𝟏𝟓, 𝟏𝟗
3𝑛 − 2 → 𝟏, 𝟒, 𝟕, 𝟏𝟎
𝑛2
− 𝑛 → 𝟎, 𝟐, 𝟔, 𝟏𝟐
2𝑛
+ 3𝑛
→ 𝟓, 𝟏𝟑, 𝟑𝟓, 𝟗𝟕
?
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?
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9. Exercise 1
In a sequence, each term after the first is the
sum of the squares of the digits of the previous
term. Thus if the first term were 12, the second
term would be 12 + 22 = 5, the third term 52 =
25, the fourth term 22 + 52 = 29, and so on. Find
the first five terms of the sequence whose first
term is 25. 25, 29, 85, 89, 145
The first three terms of a sequence are
1
4
,
1
3
,
1
2
.
The fourth term is
1
2
−
1
3
+
1
4
; henceforth, each
new term is calculated by taking the previous
term, subtracting the term before that, and then
adding the term before that.
Write down the first six terms of the sequence,
giving your answers as simplified fractions.
𝟏
𝟒
,
𝟏
𝟑
,
𝟏
𝟐
,
𝟓
𝟏𝟐
,
𝟏
𝟒
,
𝟏
𝟑
[JMO 2010 B1] In a sequence of six numbers,
every term after the second term is the sum of
the previous two terms. Also, the last term is
four times the first term, and the sum of all six
terms is 13. What is the first term?
Solution: 𝟏
𝟏
𝟒
Find the 100th term of the sequences with
the following formulae for the 𝑛th term:
a) 8𝑛 − 3 797
b) 3 − 𝑛 -97
c) 3𝑛2
− 𝑛 + 1 29901
A sequence starts with 1. Thereafter, each
new term is formed by adding all the
previous terms, and then adding 1. What
are the first 6 terms? 1, 2, 4, 8, 16,
32
Find the first 4 terms of the following
sequences:
a) 𝑛 + 3 4, 5, 6, 7
b) 3𝑛
3, 9, 27, 81
c) 𝑛3
− 𝑛2
0, 4, 18, 48
d) 𝑛2
− 4𝑛 + 1 -2, -3, -2, 1
e) 𝑛! (use your calculator) 1, 2, 6, 24
The first two terms of a sequence are 1 and
2. Each of the following terms in the
sequence is the sum of all the terms which
come before it in the sequence. Which of
these is not a term in the sequence?
A 6 B 24 C 48 D 72 E 96
(Hint: perhaps represent
the first two terms
algebraically?)
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1
2
3
4
5
6
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10. Picture Sequence Puzzle…
What are the next two pictures in this
sequence?
It’s the numbers 1, 2, 3,
… but reflected. Sneaky!
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11. Linear Sequences
What sequence does 5𝑛 give?
𝟓, 𝟏𝟎, 𝟏𝟓, 𝟐𝟎, …
What therefore would 5𝑛 − 4 give?
𝟏, 𝟔, 𝟏𝟏, 𝟏𝟔, …
What do you notice about the difference between terms in
this sequence?
It goes up by 5 each time.
What therefore do you think would be the
difference between terms for:
6𝑛 + 2 → 6
𝑛 − 1 → 1
10𝑛 − 3 → 10
3 − 𝑛 → −1
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Today’s title

12. Finding 𝑛th term formula for linear sequences
Find the 𝑛th term of the following sequence:
5, 9, 13, 17, 21 …
4𝑛 + 1
? ?
We saw that the number on
front of the 𝑛 gives us the
(first) difference between
terms.
If we had 4𝑛 as our formula,
this would give us the 4
times table. So what
‘correction’ is needed?
Bro Side Note: Why do you think this is known as a ‘linear’
sequence?
If you plotted each position with the term on some axes (e.g.
for this sequence (1,5),(2,9),(3,13),(4,17), …, it would form a
straight line. The word ‘linear’ means ‘straight’.
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13. More examples
7, 12, 17, 22, 27, … → 𝟓𝒏 + 𝟐
5, 7, 9, 11, 13, … → 𝟐𝒏 + 𝟑
2, 5, 8, 11, 14, … → 𝟑𝒏 − 𝟏
4, 10, 16, 22, 28, … → 𝟔𝒏 − 𝟐
10, 8, 6, 4, 2, … → −𝟐𝒏 + 𝟏𝟐 (or 𝟏𝟐 −
𝟐𝒏)
Quickfire Questions:
5, 8, 11, 14, 17, … → 3𝑛 + 2
3, 9, 15, 21, 27, … → 6𝑛 − 3
9, 14, 19, 24, 29, … → 5𝑛 + 4
2, 9, 16, 23, 30, … → 7𝑛 − 5
100th
term:
𝒏th term:
302
597
504
695
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? ?
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14. Test Your Understanding
10, 18, 26, 34, … → 8𝑛 + 2
2, 8, 14, 20, 26, … → 6𝑛 − 4
10, 9, 8, 7, 6, … → 11 − 𝑛
3
1
2
, 5, 6
1
2
, 8, … →
3
2
𝑛 + 2
100th
term:
𝒏th term:
802
596
−89
152
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15. Is a number in the sequence?
Is the number 598 in the sequence with 𝑛th term 3𝑛 − 2?
Could we obtain 598 using the 𝟑𝒏 − 𝟐 formula?
Yes! Working backwards, we see 𝒏 = 𝟐𝟎𝟎. So
598 is the 200th term in the sequence.
Is the number 268 in the sequence with 𝑛th term 4𝑛 − 2?
No. 𝟒𝒏 − 𝟐 = 𝟐𝟔𝟖
But adding 2 we get 270, and 270 is not divisible
by 4.
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16. Exercise 2
Find the 𝑛th term and the 300th term of the
following sequences.
5, 8, 11, 14, … → 𝟑𝒏 + 𝟐, 𝟗𝟎𝟐
4, 11, 18, 25, … → 𝟕𝒏 − 𝟑, 𝟐𝟎𝟗𝟕
11, 16, 21, 26, … → 𝟓𝒏 + 𝟔, 𝟏𝟓𝟎𝟔
6, 17,28,39, … → 𝟏𝟏𝒏 − 𝟓, 𝟑𝟐𝟗𝟓
16,20,24,28, … → 𝟒𝒏 + 𝟏𝟐, 𝟏𝟐𝟏𝟐
9,32,55,78, … → 𝟐𝟑𝒏 − 𝟏𝟒, 𝟔𝟖𝟖𝟔
1, 1
1
2
, 2, 2
1
2
, … →
𝟏
𝟐
𝒏 +
𝟏
𝟐
, 𝟏𝟓𝟎
𝟏
𝟐
Determine (with working) whether the following
numbers are in the sequence with the 𝑛th term
formula. If so, indicate the position of the term.
30 in 5𝑛 Yes (6th term)
90 in 3𝑛 + 2 No
184 in 6𝑛 − 2 Yes (31st term)
148 in 𝑛2
+ 2 No
Find the missing numbers in these linear
sequences.
3, ? , ? , ? , 19 𝟕, 𝟏𝟏, 𝟏𝟓
4, ? , ? , ? , ? , 10 (𝟓. 𝟐, 𝟔. 𝟒, 𝟕. 𝟔, 𝟖. 𝟖)
Find the formula for the 𝑛th term of the
following sequences.
6, 5, 4, 3, 2, … 𝟕 − 𝒏
5, 2, −1, −4, … 𝟖 − 𝟑𝒏
10
1
2
, 8, 5
1
2
, 3, … 𝟏𝟑 −
𝟓
𝟐
𝒏
2
1
3
, 2
7
12
, 2
5
6
, 3
1
12
𝟏
𝟒
𝒏 +
𝟐𝟓
𝟏𝟐
The 3rd term of a linear sequence is 17.
The 45th term is 269. Determine the
formula for the 𝑛th term.
𝟔𝒏 − 𝟏
Two sequences have the formulae 3𝑛 − 1
and 7𝑛 + 2. A new sequence is formed by
the numbers which appear in both of
these sequences. Determine the formula
for the 𝑛th term.
𝟐𝟏𝒏 + 𝟐
Whatever the first number is that coincides,
we’ll see it 21 later because this is the
‘lowest common multiple’ of 3 and 7. Thus
we know the formula is of the form 𝟐𝟏𝒏 + □.
It’s then simply a case of identifying which
number this is (2). This principle is known as
the ‘Chinese Remainder Theorem’.
2
1
3
4
5
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a
b
c
d
e
f
g
a
b
c
d
a
b
a
b
c
d
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17. QQQ Time!