AE 11/09/18

Version 2.0

1 of 19

Solving Problems Using Sequences

From the National Curriculum: Solve problems

• develop their mathematical knowledge, in part through solving problems and evaluating the outcomes, including multi-step problems

• develop their use of formal mathematical knowledge to interpret and solve problems, including in financial mathematics

• begin to model situations mathematically and express the results using a range of formal mathematical representations

• select appropriate concepts, methods and techniques to apply to unfamiliar and non-routine problems.

Algebra Pupils should be taught to:

• Generate terms of a sequence from either a term-to-term or a position-to-term rule

• Recognise arithmetic sequences and find the nth term

• Recognise geometric sequences and appreciate other sequences that arise.

The UKMT Problem The first term of a sequence of positive integers is 6. The other terms in the sequence follow these two rules: (i) When a term is even then divide it by 2 to obtain the next term; (ii) When a term is odd then multiply it by 5 and subtract 1 to obtain the next term. For which values of n is the nth term equal to n? IMC 2012 Q17

2 of 19

Starting points Activity 1 These questions look at generating sequences. Write down the first four terms of each sequences defined below.

Rule Answer

1

The first term is 3. Multiply a term by 2 to get the next term.

2

The first term is 4. Halve a term to get the next term.

3

The first term is 2. Square a term to get the next term.

4

The first term is 10. Subtract 4 from a term to get the next term.

5

The first term is 8. Add 10 to a term get the next term.

6

The first term is 9. Divide a term by 3 to get the next term.

We can also define sequences algebraically. Sequences can be defined by their nth term. Write down the first four terms of each sequences defined below, starting with 𝑛 = 1 in each case.

Rule Sequence

1

𝑢𝑛 = 3𝑛 − 1

2

𝑢𝑛 = 4𝑛 + 1

3

𝑢𝑛 = 5𝑛 − 2

4

𝑢𝑛 = 𝑛2

5

𝑢𝑛 = 2 × 3𝑛

6

𝑢𝑛 = (−1)𝑛3𝑛

3 of 19

For these sequences you need to use the term before to generate the next term. This is called a term to term rule. Write down the first four terms.

Sequence Answer

1

𝑢𝑛+1 = 2𝑢𝑛 − 1 𝑢1 = 3

2

𝑢𝑛+1 = 4𝑢𝑛 + 1

𝑢1 = 1

3

𝑢𝑛+1 = 3𝑢𝑛 − 2

𝑢1 = 2

4

𝑢𝑛+1 = 𝑢𝑛 − 5 𝑢1 = 1

5

𝑢𝑛+1 = 5𝑢𝑛 𝑢1 = 3

6

𝑢𝑛+1 = 1 − 𝑢𝑛 𝑢1 = 1

Extension

Does the sequence generated change if you use a different first term?

Write down sequence 1 above starting with 𝑢1 = 4. Does this change the sequence?

4 of 19

Activity 2 A linear sequence is defined to be a sequence with a common difference between every term.

We can find the 𝑛𝑡ℎ term of a linear sequence using the following rule.

𝑛𝑡ℎ term 𝑢𝑛 = 𝑢1 + (𝑛 − 1)𝑑

Where d is the difference between the terms, and 𝑢1is the first term.

Write the 𝑛𝑡ℎ term of the following Linear Sequences?

Sequence 𝐧𝐭𝐡 term

1

2, 3, 4, 5, 6, ...

2

2, 6, 10, 14, 18, 22, ...

3

4, 6, 8, 10, 12, ...

4

18, 24, 30, 36, ...

5

9, 16, 23, 30, 37, 44, ...

6

15, 20, 25, 30, 35, ...

7

4, 5, 6, 7, 8, ...

8

3, 7, 10, 14, ...

9

-5, -4, −3, −2, −1, ...

10

-8, −7, −6, −5, ...

5 of 19

Activity 3 Not all sequences are linear.

In a quadratic sequence, the difference between each term increases, or decreases, at a constant rate. For example:

3, 6, 11, 18, 27, The first difference is 3 5 7 9 The second difference is 2 2 2 Whenever a sequence has a second difference of 2, it will be connected to the sequence of square numbers. We call this quadratic.

Write down the next three terms of these quadratic sequences.

Sequence Next three terms

1

0, 3, 8, 15, 24, …

2

1, 4, 9, 16, 25, …

3

10, 13, 18, 25, 34, …

A Periodic sequence (sometimes called a cycle) is a sequence for which the same terms are repeated over and over. For example:

2, 2, 2, 7, 2, 2, 2, 7, 2, 2, 2, 7, …

Write down the next four terms of these periodic sequences.

Sequence Next four terms

4

2, 2, 2, 7, 2, 2, 2, 7, 2, 2, 2, 7….

5

4, 6, 7, 6, 4, 2, 1, 2, 4, 6, 7, 6, …

6

0, 1, 4, 5, 4, 1, 0, 1, 4 …

6 of 19

In a geometric sequence each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio. For example:

2, 6, 18, 54, 162, …

This is a sequence with a first term of 2, and each subsequent term is found by multiplying the previous term by the common ratio 3.

Write down the next three terms of these geometric sequences.

Sequence Next three terms

7

3, 6, 12, 24, …

8

1, 3, 9, 27, …

9

4, 8, 16, 32, …

Write down the first four terms of these geometric sequences using the following rules.

Sequence First four terms

10

The first term is 3, the common ratio is 4.

11

The first term is 4, the common ratio is 2.

12

The first term is 20, the common ratio is 0.5.

7 of 19

Activity 4 The original problem. Generate the sequence using the rules below. Look for the values in where the term number is equal to the term number. (For example is the 2nd term = 2?) What happens to the sequence after the first 20 terms? The first term of a sequence of positive integers is 6. The other terms in the sequence follow these two rules: (i) When a term is even then divide it by 2 to obtain the next term; (ii) When a term is odd then multiply it by 5 and subtract 1 to obtain the next term. For which values of n is the nth term equal to n? IMC 2012 Q17

8 of 19

Challenge Activity 1 The Fibonacci sequence starts like this. 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89 … The next number is found by adding up the two numbers before it. Generate a Fibonacci sequences with these starting points

Sequence Next four terms

1

2, 2, 4, …

2

3, 4, 7, …

3

4, 6, 10, …

4

6, 6, 12, …

5

9, 10, 19, …

6

15, 20, 35, …

7

4, 5, 9, …

8

3, 7, 10, …

We can use Fibonacci’s original sequence to generate a famous spiral called the Fibonacci spiral.

Describe how this spiral works.

How does it relate to the Fibonacci sequence?

9 of 19

Draw your own Fibonacci spiral. Start from the red squares.

You can also make spirals by adding more than two terms Work out what is happening in this question and answer it. JMC 2016 Q23 The diagram shows the first few squares of a ‘spiral’ sequence of squares. All but the first three squares have been labelled. After the first six squares, the sequence is continued by placing the next square alongside three existing squares—the largest existing square and two others. The three smallest squares have sides of length 1. What is the side-length of the 12th square?

10 of 19

Challenge Activity 2: Letter Sequences This is an investigation to see what sequences we can generate by placing letters onto a number grid. For example T , L, or O, S, I.

In this diagram: The starting value is the value in the top left corner for the letter, For example:

For O the starting value 13 and the sum is 192. The starting value for T is 62 and the sum is 345. For S the starting value is 7.

Fill in the rest of this table.

Letter Starting Value Sum

O 13 192

T 62 345

S 7

L

I

11 of 19

Choose one letter. For this letter, work out the letter sums for the given starting values.

Starting Value

Sum

1

2

3

4

5

6

7

8

What is the sequence for the Letter totals?

Work out the nth term of your sequence.

Use your nth term to work out the total for the letter with starting value 100.

12 of 19

Extension Try drawing your letters onto these different grids. How does the grid size affect the sequence that you find?

13 of 19

Challenge Activity 3: Josephus Flavius There is nice video to introduce this problem here https://www.youtube.com/watch?v=uCsD3ZGzMgE. Josephus Flavius was a famous Jewish historian of the first century at the time of the Second Temple destruction. During the Jewish-Roman war he got trapped in a cave with a group of 40 soldiers surrounded by Romans. The legend has it that, preferring suicide to capture, the Jews decided to form a circle and, proceeding around it, to kill every alternate remaining person until no one was left. Josephus, not keen to die, quickly found the safe spot in the circle and thus stayed alive. How did he do it? Start with an example of 10 people

1 kills 2. 2 kills 4. 3 kills 6, etc.

Going round again 1 kills 3. 5 kills 7. 9 kills 1. 5 kills 7, so 5 is still alive!

Work out who stays alive for different sized groups.

Number of people

1 2 3 4 5 6 7 8 9

Who stays alive?

Number of people

10 11 12 13 14 15 16 17

Who stays alive?

What sequence have you generated?

What is the pattern for the sequence?

Josephus Flavius was in a group of 41 people. In which position did he need to stand?

There is a full lesson plan and resources for this activity available here: https://2017.integralmaths.org/course/view.php?id=113&sectionid=2603

14 of 19

Further UKMT questions: UKMT JMC 2016 Q23 UKMT IMC 2009 Q11 UKMT IMC 2015 Q14 UKMT IMC 2015 Q16

15 of 19

Answers Activity 1

Rule Answer

1

The first term is 3, multiply by 2 to get the next term.

3, 6, 12, 24

2

The first term is 4, halve to get the next term.

4, 2, 1, 0.5

3

The first term is 2, square it to get the next term.

2, 4, 16, 256

4

The first term is 10, subtract 4 to get the next term.

10, 6, 2, -2

5

The first term is 8, add 10 to get the next term.

8, 18, 28, 38

6

The first term is 9, divide by 3 to get the next term.

9, 3, 1, 1/3

Rule Sequence

1

𝑢𝑛 = 3𝑛 − 1 2, 5, 8, 11

2

𝑢𝑛 = 4𝑛 + 1 5, 9, 13, 17,

3

𝑢𝑛 = 5𝑛 − 2 3, 8, 13, 18

4

𝑢𝑛 = 𝑛2 1, 4, 9, 16

5

𝑢𝑛 = 2 × 3𝑛 6, 18, 54, 162

6

𝑢𝑛 = (−1)𝑛3𝑛 -3, 9, -27, 81

Sequence Answer

1

𝑢𝑛+1 = 2𝑢𝑛 − 1 𝑢1 = 3

3, 5, 9, 17

2

𝑢𝑛+1 = 4𝑢𝑛 + 1 𝑢1 = 1

1, 5, 21, 85, 341

3

𝑢𝑛+1 = 3𝑢𝑛 − 2

𝑢1 = 2

2, 4, 10, 28

4

𝑢𝑛+1 = 𝑢𝑛 − 5 𝑢1 = 1

1, -4, -9, -14

5

𝑢𝑛+1 = 5𝑢𝑛

𝑢1 = 3

3, 15, 75, 375

6

𝑢𝑛+1 = 1 − 𝑢𝑛

𝑢1 = 1

1, 0, 1, 0

Extension

Does the sequence generated change if you use a different first term?

Write down sequence 1 above starting with 𝑢1 = 4, Does this change the sequence?

For the first question the new sequence is 4, 7, 13, 25. Changing the first term generates a different sequence.

16 of 19

Activity 2

Sequence 𝐧𝐭𝐡 term

1

2, 3, 4, 5, 6, ... 𝑛 + 1

2

2, 6, 10, 14, 18, 22, ... 4𝑛 − 2

3

4, 6, 8, 10, 12, ... 2𝑛 + 2

4

18, 24, 30, 36, ... 6𝑛 + 12

5

9, 16, 23, 30, 37, 44, ... 7𝑛 + 2

6

15, 20, 25, 30, 35, ... 5𝑛

7

4, 5, 6, 7, 8, ... 𝑛 + 3

8

3, 7, 10, 14, ... 4𝑛 − 1

9

-5, -4, −3, −2, −1, ... −6 + 𝑛

10

-8, −7, −6, −5, ... −9 + 𝑛

Activity 3

Sequence Next three terms

1

0, 3, 8, 15, 24, … 35, 48, 63

2

1, 4, 9, 16, 25, … 36, 49, 64

3

10, 13, 18, 25, 34, … 45, 58, 73

Sequence Next four terms

4

2, 2, 2, 7, 2, 2, 2, 7, 2, 2, 2, 7…. 2, 2, 2, 7

5

4, 6, 7, 6, 4, 2, 1, 2, 4, 6, 7, 6, … 4, 2, 1, 2

6

0, 1, 4, 5, 4, 1, 0, 1, 4 … 5, 4, 1, 0

Sequence Next three terms

7

3, 6, 12, 24, … 48, 96, 192

8

1, 3, 9, 27, … 81, 243, 729

9

4, 8, 16, 32, … 64, 128, 256

Sequence First four terms

10

The first term is 3, the common ratio is 4. 3,12, 48, 192

11

The first term is 4, the common ratio is 2. 4, 8, 16, 32

12

The first term is 20, the common ratio is 0.5. 20, 10, 5, 2.5

17 of 19

Activity 4 The first term of a sequence of positive integers is 6. The other terms in the sequence follow these two rules: (i) When a term is even then divide it by 2 to obtain the next term; (ii) When a term is odd then multiply it by 5 and subtract 1 to obtain the next term. For which values of n is the nth term equal to n? IMC 2012 Q17

18 of 19

Challenge Activity 1

Sequence Next four terms

1

2, 2, 4, … 6, 10, 16, 26,

2

3, 4, 7, … 11, 18, 29, 47

3

4, 6, 10, … 16, 26, 42, 68

4

6, 6, 12, … 18, 30, 48, 78

5

9, 10, 19, … 29, 48, 77, 125

6

15, 20, 35, … 55, 90, 145, 235

7

4, 5, 9, … 14, 23, 37, 60

8

3, 7, 10, … 17, 27, 44, 71

Work out how this spiral works.

Draw square length 1, 1, then square length 2 below, then square length 3 next to it. Once you have drawn your squares, join the corners up in a spiral.

How does it relate to the Fibonacci sequence?

Each square has a side length of a number in the Fibonacci sequence.

JMC 2016 Q23

19 of 19

Challenge Activity 2: Letter Sequences

Letter Starting Value Sum

O 13 192

T 62 345

S 7 308

L 76 355

I 79 267

For your chosen letter, work out the letter sums for the given starting values.

Starting Value Sum Example for L

1 55

2 59

3 63

4 67

5 71

6 75

7 79

8 83

What is the sequence for the Letter totals?

55, 59, 63, etc.

Work out the nth term of your sequence

4n + 51

Use your nth term to work out the total for the letter with starting value 100.

451

Challenge Activity 3: Josephus Flavius

Number of people

1 2 3 4 5 6 7 8 9

Who stays alive 1 1 3 1 3 5 7 1 3

Number of people

10 11 12 13 14 15 16 17 41

Who stays alive 5 7 9 11 13 15 1 3 19

What sequence have you generated?

1, 1, 3, 1, 3, 5, 7, 1, 3, 5, 7, 9, 11, 13, 15, 1, 3, etc.

What is the pattern for the sequence?

It’s the sequence of odd numbers, until you reach a term that is a power of two, then it starts again.

Josephus Flavius was in a group of 41 people, which position did he need to stand in?

19