Before the swap, what is the value of the fourth term?

3 The fourth position contains 3; its position number is not its value.

Solving for a number's position gives 94.6. What does that tell you?

It is not a term. A position must be a positive whole number.

Which value must receive the next addition to find the third term?

4 The previous term is 4; the requested position is not the input.

How many additions of the common difference take you from the first term to the fourth?

Three The first term needs no addition.

What fare belongs in the first position of this sequence?

₹240 The first kilometre adds ₹40 to the ₹200 booking fee.

What must you do to the rectangle's count to find one triangular total?

Halve it. The rectangle contains two copies of the same triangular total.

Which number is cubed when calculating this fourth term?

2 There are three factors of 2, but only one starting factor of 3.

After any finite number of exact halvings, what remains of a positive value?

A positive value Each multiplication uses a positive fraction, so none produces zero.

Teach Predicting What Comes Next: Exploring Sequences and Progressions in eight steps.

For each idea: show it, ask before you tell, then say one line.

Gather paper, a pencil and 13 unlit matchsticks.

Common misconceptions & worked visual for this chapter · 15 practice questions with answers

Why it matters

Equal savings deposits make the saved total rise in regular steps.
A fixed fee plus a distance charge makes longer taxi rides cost more.
Doubling a recipe doubles each ingredient, helping prepare a larger batch.

What to teach

How to teach it

Sequences and their rules

Two tables show positions unchanged while the middle values exchange places.

1. Sequences and terms

A sequence lists numbers in order; a subscript identifies a term's position.

Show: Cover the diagram's lower table with paper. Find position 2 above its value in the upper table. Read t subscript 2 as the value at position 2.
Ask: After the middle entries swap, which value will position 2 identify?
Say: It becomes 1 because position 2 now holds 1.
Used in: Numbered delivery stops keep addresses in the intended visiting order.

If your child says…“The little 2 means the value is 2.”

Ask: Before the swap, which value sits below position 2?
What’s true: The subscript chooses the position; the entry there supplies the value.

$u_n = 2n-1$

$u_4 = 2\times4-1$

$u_4 = 8-1$

2. Explicit Rule for a Sequence

Calculate a term directly from its position.

Show: Say: “The term u subscript n comes from doubling position n, then subtracting one.” Then write these lines.
Ask: What is the fourth term?
Say: The fourth term is 7, found directly from its position.
Used in: Regular seating layouts let planners calculate a later row's capacity without counting earlier rows.

If your child says…“A number can be the 94.6th term.”

Ask: Which entry would you point to when counting reaches 94.6?
What’s true: Positions count entries, so fractional positions cannot name terms.

$t_2 = 1+3$

$t_2 = 4$

$t_3 = 4+3$

3. Recursive Rule for a Sequence

Build terms from earlier values, beginning with enough starting values.

Show: Say: “Start at 1 and add 3 to each previous value.” Then write these lines. Point to 4 as it enters the next calculation.
Ask: What is the third term?
Say: The third term is 7 because the previous value, 4, gains 3.
Used in: Running expense totals use the previous total, helping track spending.

If your child says…“For term three, I add 3 to the position number.”

Ask: Which value did we find for the second term?
What’s true: The rule uses the previous term's value, starting from the given first value.

Arithmetic progressions and sums

Four adjoining matchstick squares sharing their connecting sides.

4. Arithmetic Progressions

Add the same amount at every move, including negative amounts.

Show: Build one matchstick square. Extend it to four adjoining squares, sharing sides. Record 4, 7, 10 and 13 sticks. Subtract each earlier count from the next.
Ask: For any row length, how many additions come after the first square?
Say: There is one fewer addition than squares.
Used in: Equal daily savings help plan when a target amount will be reached.

If your child says…“The fourth term needs four additions.”

Ask: How many extensions did we make after building the first square?
What’s true: The first term is already present, so reaching the fourth takes three additions.

$t_3 = 240+(3-1)\times40$

$t_3 = 240+2\times40$

$t_3 = 240+80$

5. Arithmetic Progressions: taxi fare

Define what each term measures before counting changes.

Show: Say: “This example charges ₹200 for booking and ₹40 per kilometre. Terms record fares after each kilometre, so the first is ₹240.” Then write these lines.
Ask: What is the fare after three kilometres?
Say: The fare is ₹320, including two increases after the first kilometre's fare.
Used in: Delivery charges can combine fixed and distance charges, helping estimate costs.

If your child says…“The booking fee is the first term.”

Ask: Has the taxi covered a kilometre when we record the first term?
What’s true: The first listed fare already includes one kilometre's charge.

Two coloured dot staircases combine into a rectangle with four rows and five columns.

6. Sum of the First n Natural Numbers

Two matching staircases reveal the total by forming a rectangle.

Show: Find both staircases in the diagram, then their combined rectangle.
Ask: How many dots does one copy contain?
Say: One copy has 10 dots, half the rectangle.
Used in: Triangular stacking totals help count objects without counting every object separately.

If your child says…“One triangle contains the whole rectangle's total.”

Ask: How many copies of the original staircase fill the rectangle?
What’s true: The rectangle holds two equal triangular totals, so either triangle contains half.

Geometric progressions

$t_4 = 3\times2\times2\times2$

$t_4 = 3\times2^3$

$t_4 = 3\times8$

7. Geometric Progressions

Multiply by the same ratio at every move.

Show: Write 3, 6, 12. Divide each term by the one before it to check the multiplier, 2. Write these lines, saying: “Double three times after the starting 3.”
Ask: What is the fourth term?
Say: The fourth term is 24, with the starting 3 used once.
Used in: Repeated recipe doubling helps calculate ingredient amounts for larger batches.

If your child says…“I cube both the starting 3 and the multiplier 2.”

Ask: How many times does 3 appear in the expanded first line?
What’s true: Only the multiplier repeats, so only it receives the power.

A halving table beside separate graph points approaching the horizontal axis.

8. Visualising a GP

A ratio between zero and one makes positive terms approach zero.

Show: Look at the diagram's table and follow the halving arrows. Match each position and value to its graph point. Compare the drops between neighbouring points.
Ask: If these values keep halving, will any later point reach zero?
Say: The points approach zero without reaching it: halving a positive value always leaves something positive. Their shrinking drops keep these points from lying on a straight line.
Used in: Halving paper makes equal pieces for sharing.

If your child says…“Enough halvings will make the value exactly zero.”

Ask: What happens when you halve any positive value that still remains?
What’s true: Every finite number of exact halvings leaves a positive value.

A real problem

Logarithm tables made multiplication easier.

Napier published his explanation of logarithms in 1614.

His method paired an increasing arithmetic progression with a decreasing geometric progression.

The tables replaced laborious multiplication with simpler addition.

University of St Andrews Brown University Library Open University

Open logarithm-table book beside a pencil and handwritten calculations.

Concept Lab · Common misconception

The tempting mistake in Predicting What Comes Next: Exploring Sequences and Progressions

Open standalone misconception page →

Does an increasing sequence always add the same number?

Wrong

2, 4, 8, 16 is arithmetic because every term is larger than the previous one.

Why it feels right. The regular increase looks like equal steps.

Right

An arithmetic progression has a constant difference. A geometric progression has a constant ratio.

Worked example

Term 12

Term 24

Term 38

Term 416

Term 532

Each term doubles. The rising heights make the changing differences visible.

The differences are 2, 4 and 8, while the ratios are all 2.
The terms double, so the pattern gives 32 as the next term.

Ask your child eight quick questions.

One per step. The answer and the reason are given.

Before the swap, what is the value of the fourth term?

3 The fourth position contains 3; its position number is not its value.
Solving for a number's position gives 94.6. What does that tell you?

It is not a term. A position must be a positive whole number.
Which value must receive the next addition to find the third term?

4 The previous term is 4; the requested position is not the input.
How many additions of the common difference take you from the first term to the fourth?

Three The first term needs no addition.
What fare belongs in the first position of this sequence?

₹240 The first kilometre adds ₹40 to the ₹200 booking fee.
What must you do to the rectangle's count to find one triangular total?

Halve it. The rectangle contains two copies of the same triangular total.
Which number is cubed when calculating this fourth term?

2 There are three factors of 2, but only one starting factor of 3.
After any finite number of exact halvings, what remains of a positive value?

A positive value Each multiplication uses a positive fraction, so none produces zero.

15 more questions with answers

Teach Predicting What Comes Next: Exploring Sequences and Progressions in eight steps.

Why it matters