10.1 Sequences

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The big idea. How can we be sure a pattern will keep working? Represent sequences, find general rules for arithmetic, geometric and quadratic sequences, and use recursive definitions.

Representing a sequence

Show a sequence four ways: a picture, a table of terms, a graph (points, not a joined line) and an algebraic rule.
The term number n and the term u_n: u₁ is the first term.
Triangle numbers: 1, 3, 6, 10, 15 ... rule u_n = n(n + 1) ÷ 2. The graph curves upwards.
A straight-line graph means a constant difference (arithmetic); a curve means something else.

Shifting a known pattern

Name the base sequence: squares (1, 4, 9, 16), cubes (1, 8, 27, 64), triangle numbers, powers of 2.
Find the constant shift: 0, 3, 8, 15, 24 is one less than the squares, so u_n = n² − 1.
Check with a term you did not use: u₆ = 36 − 1 = 35.

Quadratic sequences (second differences)

Build a difference table. If the first differences are not constant but the second differences are, the rule is quadratic: u_n = an² + bn + c.
a = second difference ÷ 2. Subtract an² from each term to leave a linear sequence bn + c.
Example 4, 9, 16, 25 ... first differences 5, 7, 9; second differences 2, 2, so a = 1; u_n = n² + 2n + 1.
Check the rule with a term you did not use.

Arithmetic sequences

Constant common difference d: 5, 8, 11, 14 has d = 3.
u_n = u₁ + (n − 1)d. For 5, 8, 11: u_n = 5 + 3(n − 1) = 3n + 2.
Find a given term: u₂₀ = 5 + 19 × 3 = 62.
Find which term equals a value: 3n + 2 = 50 gives n = 16.

Geometric sequences

Constant common ratio r: 3, 6, 12, 24 has r = 2.
u_n = u₁ × rⁿ⁻¹. For 3, 6, 12: u_n = 3 × 2ⁿ⁻¹.
A negative r alternates the sign: 2, −6, 18, −54 has r = −3.
u₆ = 3 × 2⁵ = 96.

Working backwards and equal properties

Given two terms, count the gap. If u₃ = 11 and u₇ = 27: 4 steps give 16, so d = 4; u₁ = 11 − 8 = 3.
Geometric: if u₂ = 6 and u₅ = 48, then r³ = 8, r = 2 and u₁ = 3.
Equal differences: if x, x + 4, 3x are arithmetic then (x + 4) − x = 3x − (x + 4), so 4 = 2x − 4 and x = 4.
Equal ratios: if a, 6, 24 are geometric then 6 ÷ a = 24 ÷ 6 = 4, so a = 1.5.
Test each solution by writing out the terms.

Recursive definitions

A recursive definition gives the first term and a rule that uses the previous term.
Arithmetic: u₁ = 5, u_n+1 = u_n + 3. Geometric: u₁ = 3, u_n+1 = 2u_n.
The explicit rule gives any term directly: u_n = 3n + 2. Convert between them.
Always state the starting value and use the precise words.

Common mistakes

✕ “Join the points of a sequence graph with a line.” ✓ Only whole-number terms exist, so plot separate points.
✕ “u_n = u₁ + nd.” ✓ The first term has no difference added: u_n = u₁ + (n − 1)d.
✕ “Geometric rule: u_n = u₁ × rⁿ.” ✓ The power is n − 1: u_n = u₁ × rⁿ⁻¹.
✕ “a in an² + bn + c is the second difference.” ✓ a is half the second difference.
✕ “A recursive rule alone is complete.” ✓ You must also state the starting value.

Practice booklet

Unit 10.1 Practice Booklet

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Practice Booklet

MYP 4 · Year 10 Mathematics (Extended)

Unit 10.1 · Sequences

Name:

10.1.1: Representing sequences · Questions 1–15
10.1.2: Shifted sequences · Questions 16–30
10.1.2b: Second differences · Questions 31–45
10.1.3: Arithmetic sequences · Questions 46–60
10.1.4: Geometric sequences · Questions 61–75
10.1.5: Working backwards · Questions 76–90
10.1.6: Equal differences and equal ratios · Questions 91–105
10.1.7: Recursive rules · Questions 106–120
Homework: Homework page and checking record
Answers: For checking and correcting in green pen

MYP 4 · Year 10 · Unit 10.1 Sequences · Practice BookletMs B’s mathscafe · page 1

How to use this booklet and your homework

Read this first

Choosing your level

Every part of the unit has four levels. Start where you feel comfortable and move between them freely. Most questions ask you to explain as well as answer, because being able to say why is what makes a method stick.

Learning

Small numbers, with counters or a diagram beside you.

Practising

The standard question, with your reasoning written out.

Challenging

Working backwards, and finding what is wrong with someone else's thinking.

Mastery

Explaining why a rule must be true, and testing it on new cases.

Working on your homework

Pens and setting out

Answers in blue or black non-erasable pen, or in pencil.
Colour and highlighters are for annotating your work and showing the steps of a method.
Never erase a mistake. Rework it underneath so you can see what changed.

Green pen is for checking. When you finish, use the answers at the back to check your work.

Got it wrong? Circle the question number, then make your correction and write your error analysis in green pen.
Still can’t work out how to solve it? Put a star ★ next to the question number, so we know to go over it in class.

Each sheet

Your name goes on every sheet, because they are handed in one at a time.
Each question is worth up to 3 marks. If a question has parts (a, b, c…), each part is marked out of 3. At the end of each level, write your score out of the total for that level.
Questions are numbered 1, 2, 3… all the way through the booklet, so we can say exactly which ones to do.
If you are stuck after trying, ask for help. Showing me what you tried is a good start.

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10.1.1 Representing sequences

Learning

Name:

Key vocabulary

Word	Meaning	Example
sequence	a list of numbers that follows a pattern	1, 3, 6, 10, …
term	one number in the sequence	the 3rd term is 6
term number, n	the position of a term in the list	n = 4 is the 4th term
u_n	the term in position n	u₄ = 10
rule	a formula that gives any term from its term number	u_n = n(n + 1) ÷ 2

One sequence can be shown four ways: a picture, a table, a graph and a rule.

Learning

The first four terms of a sequence are 1, 3, 6, 10.
a) Work out the gaps between the terms.
b) Write the next two terms.

A sequence has the rule u_n = 3n + 1. Work out u₁, u₂ and u₃.

The points (1, 2), (2, 4), (3, 6), (4, 8) are plotted on a graph.
a) Write the sequence.
b) Do the points lie on a straight line? Explain.
c) Write the rule u_n = ____

Staircase pictures: step 1 has 1 square, step 2 has 4 squares (a 2 by 2 block), step 3 has 9 squares. Draw step 4, count its squares and write the rule.

Learning score / 21

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10.1.1 Representing sequences

Practising

Name:

Practising

The triangle numbers have the rule u_n = n(n + 1)2. Work out
a) u₈
b) u₁₀

Complete the table for u_n = n² + 3 with n = 1 to 5.
a) Complete the table.
b) Do the points lie on a straight line? Explain.

A sequence goes 5, 8, 11, 14, 17.
a) Find the gaps.
b) Will the graph be a straight line or a curve? Explain.
c) Write a rule for u_n.

Is 45 a triangle number? Solve n(n + 1) = 2 × 45 by looking for two numbers next to each other with product 90.

Practising score / 24

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10.1.1 Representing sequences

Challenging · Mastery

Name:

Challenging

Compare 2, 6, 12, 20, 30 with the triangle numbers 1, 3, 6, 10, 15. Write the rule for u_n.

Find and describe the mistake. Dev says the sequence 1, 3, 6, 10, 15 has the rule u_n = 2n − 1 because u₁ = 1 and u₂ = 3.

Step 11 of the triangle pattern has 66 counters and step 12 has 78. How many counters are added, and why does that number make sense in the picture?

Is 100 a triangle number? Explain using n(n + 1) = 200.

Challenging score / 12

Mastery

Draw two copies of step 5 of the triangle pattern fitted together to make a rectangle. Use it to explain why u₅ = 5 × 62.

Sam says that a table of five terms proves a rule. Explain why it does not, and say how the rectangle picture does prove the rule for every n.

A graph of a sequence has points that curve upwards. Write the first five terms of one possible sequence and explain how you know the graph curves without plotting it.

Mastery score / 9

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10.1.2 Shifted sequences

Learning

Name:

Name it, shift it, check it

Step	What you do	Example: 2, 5, 10, 17, 26
Name it	Which known sequence is it close to? Squares or cubes	1, 4, 9, 16, 25
Shift it	Subtract. What constant is added or taken away every time?	always + 1
Check it	Substitute a term you did not use	n = 5: 25 + 1 = 26

Squares: 1, 4, 9, 16, 25. Cubes: 1, 8, 27, 64, 125.

Learning

Write the first five square numbers and the first five cube numbers.

Compare 0, 3, 8, 15, 24 with the squares 1, 4, 9, 16, 25.
a) What is added to each square to give the term?
b) Complete the rule u_n = n² ____

A sequence has the rule u_n = n² + 2. Work out u₁, u₂, u₃ and u₄.

Name it, shift it: find the rule for 2, 5, 10, 17, 26.

Learning score / 15

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10.1.2 Shifted sequences

Practising

Name:

Practising

Find the rule for
a) 5, 8, 13, 20, 29
b) −1, 2, 7, 14, 23

Find the rule for
a) 0, 7, 26, 63, 124
b) 2, 9, 28, 65, 126 (compare with the cubes)

A sequence has the rule u_n = n² − 3.
a) Find u₈.
b) Which term equals 97?
c) Is 45 a term? Explain.

The terms 3, 6, 11, 18 were used to find the rule n² + 2. Check the rule with the fifth term, 27.

Practising score / 24

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10.1.2 Shifted sequences

Challenging · Mastery

Name:

Challenging

Find the rule for 3, 10, 29, 66, 127.

Find and describe the mistake. Ali says the rule for 4, 7, 12, 19 is n² + 4 because the first term is 4.

a) Find the rule for −4, −1, 4, 11, 20. b) Which term equals 76?

The sequence 10, 17, 36, 73, 134 is the cubes shifted by a constant. Write the rule and find u₁₀.

Challenging score / 15

Mastery

Why is it important to test a term you did not use when you build a rule? Explain.

How can you tell from the terms of a sequence whether to compare it with the squares or with the cubes? Use two examples.

Make your own sequence by shifting the squares by a number you choose. Write the first five terms and the rule, then swap with a partner to check.

Mastery score / 9

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10.1.2b Second differences

Learning

Name:

Key vocabulary

Word	Meaning	Example: 2, 6, 12, 20, 30
first difference	the gap between one term and the next	4, 6, 8, 10
second difference	the gap between one first difference and the next	2, 2, 2
quadratic sequence	a sequence whose second differences are constant	rule an² + bn + c

a is half of the second difference. Then b = (first gap) − 3a, and c = u₁ − a − b.

Learning

Find the first differences and the second differences of 2, 6, 12, 20, 30.

Find the first and second differences of 1, 4, 9, 16, 25. Is the sequence quadratic?

Each sequence has a constant second difference. Find a, which is half of the second difference.
a) second difference 2
b) second difference 6
c) second difference 10

For u_n = n² + 2n find u₁ to u₄. Then check that the second differences are constant.

Learning score / 18

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10.1.2b Second differences

Practising

Name:

Practising

Find the rule for
a) 6, 9, 14, 21
b) 2, 7, 14, 23

Find the rule for 3, 10, 21, 36, 55.

Find the rule for 5, 4, 5, 8, 13.

A sequence has the rule u_n = 2n² − 3n.
a) Find u₁₀.
b) Which term equals 20? Try values of n.

Practising score / 18

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10.1.2b Second differences

Challenging · Mastery

Name:

Challenging

Find the rule for 7, 18, 33, 52, 75.

Find and describe the mistake. Mia says that the second difference of 3, 8, 15, 24 is 2, so a = 2.

The sequence 5, 11, 21, 35, 53 is quadratic. Find the rule, then find u₂₀.

The first differences of a quadratic sequence start 5, 9, 13 and its first term is 3. Write the first four terms and find the rule.

Challenging score / 12

Mastery

Explain why the second difference of an² + bn + c is 2a. Use the terms for n = 1, 2, 3.

Explain why 2, 4, 8, 16, 32 is not a quadratic sequence.

A sequence has the rule u_n = n² + bn and u₂ = 10. Find b and write the first four terms.

Mastery score / 9

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10.1.3 Arithmetic sequences

Learning

Name:

Key vocabulary

Word	Meaning	Example: 4, 9, 14, 19
arithmetic sequence	the same amount is added every time	+ 5 each time
common difference, d	the amount added (negative if the terms fall)	d = 5
u₁	the first term	u₁ = 4
rule	u_n = u₁ + (n − 1)d	4 + (n − 1) × 5 = 5n − 1

Learning

Find d for
a) 4, 9, 14, 19
b) 50, 42, 34, 26

An arithmetic sequence has u₁ = 6 and d = 4. Write u₂ to u₅.

Use u_n = u₁ + (n − 1)d to write the rule for 3, 7, 11, 15.

An arithmetic sequence has the rule u_n = 5n − 2. Write
a) u₁
b) d
c) u₁₀

Learning score / 21

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10.1.3 Arithmetic sequences

Practising

Name:

Practising

Find the rule for 8, 15, 22, 29 and use it to find u₄₀.

Find the rule for 90, 84, 78, 72 and use it to find u₁₈.

An arithmetic sequence has u₁ = 1.5 and d = 0.25. Find u₁₂.

Which term of 5, 8, 11, 14, … equals 101?

Practising score / 12

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10.1.3 Arithmetic sequences

Challenging · Mastery

Name:

Challenging

An arithmetic sequence has u₁ = −4 and d = −1.5. Find u_n and u₉.

Find and describe the mistake. Ben says u₃₀ of 4, 9, 14, … is 4 + 30 × 5 = 154.

An arithmetic sequence has u₁ = 100 and d = −7. Find the first negative term.

Is 250 a term of 7, 12, 17, …? Explain.

Challenging score / 12

Mastery

Use the jumps on a number line to explain why u_n = u₁ + (n − 1)d and not u₁ + nd.

A sequence has d = 0. What does its graph look like? Is it still an arithmetic sequence?

Show that u_n = 3n + 4 is arithmetic by working out u_n+1 − u_n.

Mastery score / 9

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10.1.4 Geometric sequences

Learning

Name:

Key vocabulary

Word	Meaning	Example: 3, 6, 12, 24
geometric sequence	each term is multiplied by the same number	× 2 each time
common ratio, r	the multiplier (divide a term by the one before)	r = 2
rule	u_n = u₁ × r^{n − 1}	3 × 2^{n − 1}

Learning

Find r for
a) 2, 6, 18, 54
b) 400, 200, 100, 50

A geometric sequence has u₁ = 4 and r = 3. Write u₂ to u₅.

Use u_n = u₁ × r^{n − 1} to write the rule for 5, 10, 20, 40.

A geometric sequence has u₁ = 3 and r = 2. Write u₁ to u₆.

Learning score / 15

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10.1.4 Geometric sequences

Practising

Name:

Practising

Find the rule for 3, 9, 27, 81 and use it to find u₈.

Find the rule for 1000, 500, 250, 125 and use it to find u₇.

A geometric sequence has u₁ = −2 and r = 5. Find u₄.

Which term of 2, 6, 18, 54, … equals 1458?

Practising score / 12

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10.1.4 Geometric sequences

Challenging · Mastery

Name:

Challenging

A geometric sequence has u₁ = 6 and r = −12. Find u_n and u₆.

Find and describe the mistake. Zoe says u₅ of 2, 6, 18, … is 2 × 3⁵ = 486.

Find the rule for 1, −2, 4, −8, … and say, without listing every term, whether u₁₅ is positive or negative.

£500 is invested and grows by 4% each year. The amounts form a geometric sequence with u₁ = 500 at the start. Write r and find the amount after 3 years.

Challenging score / 12

Mastery

Use the doubling bars picture to explain why u_n = u₁ × r^{n − 1} and not u₁ × rⁿ.

A sequence has r = 1. What does its graph look like? Is it also an arithmetic sequence? Explain.

A geometric sequence has 0 < r < 1 and u₁ > 0. Explain why the terms get smaller but never reach 0.

Mastery score / 9

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10.1.5 Working backwards

Learning · Practising

Name:

Working backwards

Sequence	Jump between two terms	Example
arithmetic	from u_a to u_b there are (b − a) jumps, so d = (difference of terms) ÷ (b − a)	u₃ = 8, u₆ = 23: d = 15 ÷ 3 = 5
geometric	r^{b − a} = u_b ÷ u_a, then take the root	u₂ = 6, u₅ = 162: r³ = 27

Learning

An arithmetic sequence has u₁ = 6 and u₂ = 11. Find d and u₃.

A geometric sequence has u₁ = 5 and u₂ = 15. Find r and u₃.

An arithmetic sequence has u₃ = 14 and d = 4. Work back to find u₁ (subtract d twice).

A geometric sequence has u₃ = 36 and r = 3. Work back to find u₁ (divide by r twice).

Learning score / 12

Practising

An arithmetic sequence has u₃ = 8 and u₆ = 23. Find d and u₁.

A geometric sequence has u₂ = 6 and u₅ = 162. Find r and u₁.

An arithmetic sequence has the rule u_n = 7n − 4. Which term equals 87?

A geometric sequence has the rule u_n = 2 × 3^{n − 1}. Which term equals 486?

Practising score / 12

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10.1.5 Working backwards

Challenging · Mastery

Name:

Challenging

A geometric sequence has u₄ = 3 and u₇ = 6. Find r exactly, without rounding.

An arithmetic sequence has u₂ = 15 and u₅ = −3. Find d and u₁.

Find and describe the mistake. Kai says an arithmetic sequence with u₂ = 7 and u₆ = 23 has d = (23 − 7) ÷ 6.

An arithmetic sequence has u₁ = 5, d = 4 and u_n = 77. How many terms are there up to u_n?

Challenging score / 12

Mastery

A geometric sequence has u₂ = 4 and u₆ = 64. Find both possible sequences and their u₁.

Explain why u₆ ÷ u₂ = r⁴ for a geometric sequence.

An arithmetic sequence has u₄ = 10 and u₉ = −5. Is 0 a term? Explain.

Mastery score / 9

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10.1.6 Equal differences and equal ratios

Learning

Name:

The two properties

Three consecutive terms a, b, c	What is equal	So
arithmetic	b − a = c − b (equal differences)	b is the mean of a and c
geometric	b ÷ a = c ÷ b (equal ratios)	b² = ac

Learning

x, x + 5, x + 10 are consecutive terms of an arithmetic sequence. Show that both differences equal 5 for any x.

4, 8, m are consecutive terms of a geometric sequence. Find m.

3, k, 11 are consecutive terms of an arithmetic sequence. Find k.

2, k, 18 are consecutive terms of a geometric sequence with k > 0. Find k.

Learning score / 12

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10.1.6 Equal differences and equal ratios

Practising

Name:

Practising

n, 3n − 2, 4n + 1 are consecutive terms of an arithmetic sequence. Find n and the three terms.

5, k, 45 are consecutive terms of a geometric sequence. Find both possible values of k and the sequence each gives.

x + 2, 2x + 1, 4x − 3 are consecutive terms of an arithmetic sequence. Find x and the terms.

x, x + 4, x + 10 are consecutive terms of a geometric sequence. Find x and the common ratio.

Practising score / 12

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10.1.6 Equal differences and equal ratios

Challenging · Mastery

Name:

Challenging

n + 1, 2n + 3, 4n are consecutive terms of an arithmetic sequence. Find n and the terms.

100

Find and describe the mistake. To find k in 3, k, 27 (geometric), Ravi writes k − 3 = 27 − k, so k = 15.

101

x − 1, x + 1, 2x + 2 are consecutive terms of a geometric sequence. Show that x² − 2x − 3 = 0, solve it, and say which value gives a valid geometric sequence.

102

Three numbers in an arithmetic sequence add to 24 and the largest is 3 times the smallest. Find them.

Challenging score / 12

Mastery

103

Explain why the middle term of three consecutive terms of an arithmetic sequence is the mean of the other two.

104

Explain why, for three positive consecutive terms of a geometric sequence, the middle term is the square root of the product of the other two.

105

Can −5, k, −20 be consecutive terms of a geometric sequence? Find all possible values of k.

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10.1.7 Recursive rules

Learning

Name:

Key vocabulary

Word	Meaning	Example
recursive rule	gives the first term and how to get each term from the one before	u₁ = 5, u_n+1 = u_n + 3
explicit rule	gives any term straight from n	u_n = 3n + 2

Arithmetic: u_n+1 = u_n + d. Geometric: u_n+1 = r × u_n. Always state u₁.

Learning

106

A sequence is defined by u₁ = 5 and u_n+1 = u_n + 3. Write u₂, u₃ and u₄.

107

A sequence is defined by u₁ = 2 and u_n+1 = 3u_n. Write u₂, u₃ and u₄.

108

Say in words what u_n+1 = u_n − 4 means.

109

The sequence 20, 17, 14, 11 is arithmetic. Write its recursive rule.

Learning score / 12

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10.1.7 Recursive rules

Practising

Name:

Practising

110

A sequence has u₁ = 4 and u_n+1 = u_n + 7. Find u₂ to u₅, then write the explicit rule.

111

A sequence has u₁ = 100 and u_n+1 = 0.5u_n. Find u₂ to u₅, then write the explicit rule.

112

Write a recursive rule for
a) u_n = 8 − 2n
b) u_n = 3 × 2^{n − 1}

113

A sequence has u₁ = 6 and u_n+1 = −3u_n. Write the explicit rule.

Practising score / 15

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10.1.7 Recursive rules

Challenging · Mastery

Name:

Challenging

114

A sequence has u₁ = 2 and u_n+1 = 2u_n + 1. Find u₂ to u₅. Is the sequence arithmetic, geometric or neither? Explain.

115

Find and describe the mistake. Tom writes the sequence 5, 9, 13, 17 as u₁ = 5 and u_n+1 = 4u_n.

116

A sequence has u₁ = 80 and u_n+1 = 1.1u_n. Find the first term that is greater than 100.

117

An arithmetic sequence has u₁ = 7 and u₄ = 22. Write its recursive rule and its explicit rule.

Challenging score / 12

Mastery

118

A sequence has u₁ = 3 and u_n+1 = u_n + 2n. Find u₂ to u₅ and explain why no single d fits.

119

Give one advantage of an explicit rule and one advantage of a recursive rule.

120

For 5, 10, 20, 40, … write the recursive and the explicit rule. Which is quicker for finding u₃₀, and why?

Mastery score / 9

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Homework and checking record

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Sheet	Questions	Which ones I did	Checked in green ✓	Teacher
10.1.1 Representing sequences	Q1–15
10.1.2 Shifted sequences	Q16–30
10.1.2b Second differences	Q31–45
10.1.3 Arithmetic sequences	Q46–60
10.1.4 Geometric sequences	Q61–75
10.1.5 Working backwards	Q76–90
10.1.6 Equal differences and equal ratios	Q91–105
10.1.7 Recursive rules	Q106–120

Error analysis

Circle the number of any question you got wrong. Star ★ it if you still cannot solve it after correcting. Then write the questions you need to rework, and what you will do differently:

One thing I understand better now

Challenging

1145, 11, 23, 47. The differences 3, 6, 12 are not constant and the ratios 2.5, 2.2, 2.09… are not constant, so it is neither1154u_n multiplies each term by 4, which gives 5, 20, 80. The sequence adds 4 each time, so the rule is u_n+1 = u_n + 4116u₂ = 88, u₃ = 96.8, u₄ = 106.48. The first term greater than 100 is u₄117d = 15 ÷ 3 = 5. Recursive: u₁ = 7 and u_n+1 = u_n + 5. Explicit: u_n = 5n + 2

Mastery

1185, 9, 15, 23. The amounts added are 2, 4, 6, 8, so the difference changes and there is no constant d119Explicit: you can find any term, such as u₁₀₀, directly. Recursive: it shows how the sequence grows from term to term and is easy to enter in a calculator120Recursive: u₁ = 5 and u_n+1 = 2u_n. Explicit: u_n = 5 × 2^{n − 1}. The explicit rule is quicker: u₃₀ = 5 × 2²⁹ in one step, instead of 29 repeated doublings

Revision slides as text (easy to translate)

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Slide 1: Sequences

Revision slides: everything you need for the test, one idea per slide.

Slide 2: Representing a sequence

Show a sequence four ways: a picture, a table of terms, a graph (points, not a joined line) and an algebraic rule.

The term number n and the term u_n: u₁ is the first term.

Triangle numbers: 1, 3, 6, 10, 15 ... rule u_n = n(n + 1) ÷ 2. The graph curves upwards.

A straight-line graph means a constant difference (arithmetic); a curve means something else.

Slide 3: Shifting a known pattern

Name the base sequence: squares (1, 4, 9, 16), cubes (1, 8, 27, 64), triangle numbers, powers of 2.

Find the constant shift: 0, 3, 8, 15, 24 is one less than the squares, so u_n = n² − 1.

Check with a term you did not use: u₆ = 36 − 1 = 35.

Slide 4: Quadratic sequences (second differences)

Build a difference table. If the first differences are not constant but the second differences are, the rule is quadratic: u_n = an² + bn + c.

a = second difference ÷ 2. Subtract an² from each term to leave a linear sequence bn + c.

Example 4, 9, 16, 25 ... first differences 5, 7, 9; second differences 2, 2, so a = 1; u_n = n² + 2n + 1.

Check the rule with a term you did not use.

Slide 5: Arithmetic sequences

Constant common difference d: 5, 8, 11, 14 has d = 3.

u_n = u₁ + (n − 1)d. For 5, 8, 11: u_n = 5 + 3(n − 1) = 3n + 2.

Find a given term: u₂₀ = 5 + 19 × 3 = 62.

Find which term equals a value: 3n + 2 = 50 gives n = 16.

Slide 6: Geometric sequences

Constant common ratio r: 3, 6, 12, 24 has r = 2.

u_n = u₁ × rⁿ⁻¹. For 3, 6, 12: u_n = 3 × 2ⁿ⁻¹.

A negative r alternates the sign: 2, −6, 18, −54 has r = −3.

u₆ = 3 × 2⁵ = 96.

Slide 7: Working backwards and equal properties

Given two terms, count the gap. If u₃ = 11 and u₇ = 27: 4 steps give 16, so d = 4; u₁ = 11 − 8 = 3.

Geometric: if u₂ = 6 and u₅ = 48, then r³ = 8, r = 2 and u₁ = 3.

Equal differences: if x, x + 4, 3x are arithmetic then (x + 4) − x = 3x − (x + 4), so 4 = 2x − 4 and x = 4.

Equal ratios: if a, 6, 24 are geometric then 6 ÷ a = 24 ÷ 6 = 4, so a = 1.5.

Test each solution by writing out the terms.

Slide 8: Recursive definitions

A recursive definition gives the first term and a rule that uses the previous term.

Arithmetic: u₁ = 5, u_n+1 = u_n + 3. Geometric: u₁ = 3, u_n+1 = 2u_n.

The explicit rule gives any term directly: u_n = 3n + 2. Convert between them.

Always state the starting value and use the precise words.

Slide 9: Don’t fall for these

✕ Wrong: Join the points of a sequence graph with a line.

✓ Correct: Only whole-number terms exist, so plot separate points.

✕ Wrong: u_n = u₁ + nd.

✓ Correct: The first term has no difference added: u_n = u₁ + (n − 1)d.

✕ Wrong: Geometric rule: u_n = u₁ × rⁿ.

✓ Correct: The power is n − 1: u_n = u₁ × rⁿ⁻¹.

✕ Wrong: a in an² + bn + c is the second difference.

✓ Correct: a is half the second difference.

✕ Wrong: A recursive rule alone is complete.

✓ Correct: You must also state the starting value.

Slide 10: Test yourself

1. Which is the rule for the triangle numbers 1, 3, 6, 10?

Answer: n(n + 1) ÷ 2. n = 4 gives 4 × 5 ÷ 2 = 10.

2. The sequence 0, 3, 8, 15, 24 has the rule...

Answer: n² − 1. It is one less than the square numbers.

3. An arithmetic sequence has u₃ = 11 and u₇ = 27. What is d?

Answer: 4. Four steps add 16, so d = 4.

4. With u₃ = 11 and d = 4, what is u₁?

Answer: 3. 11 − 2 × 4 = 3.