Counting & sequences
🎯 In this topic you will
- Count forwards and backwards using fractions and decimals.
- Find and use the position-to-term rule of a sequence.
🧠 Key Words
- position
- position-to-term rule
- term
- term-to-term rule
Show Definitions
- position: The place number of a term in a sequence, such as 1st, 2nd, or 10th.
- position-to-term rule: A rule or formula that gives the value of any term directly from its position number in the sequence.
- term: One number or item in a sequence.
- term-to-term rule: A rule that tells you how to get from one term to the next term in a sequence.
🔢 From Term-to-Term to Position-to-Term
In Stage 5, you learnt how to use a term-to-term rule to find the next term in a sequence. In this unit, you will learn how to use the position-to-term rule to find any term in a sequence.
How to Work it Out: The 25th term is $25 \times 4 = 100$
❓ EXERCISES
$1$.
a. Find the position-to-term rule for the numbers in this table.
| Position | Term |
|---|---|
| $1$ | $6$ |
| $2$ | $12$ |
| $3$ | $18$ |
| $4$ | $24$ |
b. What is the $10$th term of the sequence $6, 12, 18 \ldots$?
👀 Show answer
a. The position-to-term rule is multiply by $6$, so the term is $6n$.
b. The $10$th term is $10 \times 6 = 60$.
$2$. The numbers in this sequence increase by equal amounts each time.
a. Write the three missing numbers.
$3, \square, \square, \square, 15$
b. What is the term-to-term rule for the sequence?
c. What is the position-to-term rule for the sequence?
👀 Show answer
a. The missing numbers are $6, 9, 12$.
b. The term-to-term rule is add $3$.
c. The position-to-term rule is multiply by $3$, so the term is $3n$.
$3$.
a. Follow the instructions in the flow diagram to generate a sequence.
b. What is the position-to-term rule for the sequence?
c. Imagine the sequence continues forever.
What is the $50$th term in the sequence?
👀 Show answer
a. The sequence is $8, 16, 24, 32, 40, 48$.
We stop before writing $56$ because $56$ is more than $50$.
b. The position-to-term rule is multiply by $8$, so the term is $8n$.
c. The $50$th term is $50 \times 8 = 400$.
$4$. Here is the start of a sequence using rectangles and triangles. Each rectangle is numbered.
The sequence continues in the same way.
a. How many triangles are there in the shape with $50$ rectangles?
How many rectangles and triangles are there altogether in that shape?
b. Jodi starts to make a table showing the position (shape number) and the term (total number of rectangles and triangles). Copy and complete her table.
| Position | Term |
|---|---|
| $1$ | $3$ |
| $2$ | |
| $3$ | $9$ |
| $4$ | $12$ |
| $5$ |
c. What is the position-to-term rule for Jodi's sequence?
d. What is the $50$th term in the sequence?
👀 Show answer
a. With $50$ rectangles, there are $100$ triangles (two triangles for each rectangle).
Altogether there are $50 + 100 = 150$ shapes.
b. Completed table terms: position $2 \to 6$ and position $5 \to 15$.
c. The position-to-term rule is multiply by $3$, so the term is $3n$.
d. The $50$th term is $50 \times 3 = 150$.
$5$. Pablo counts up in quarters.
What are the two missing numbers?
$\frac{1}{4}, \frac{1}{2}, \frac{3}{4}, 1, \square, 1\frac{1}{2}, \square, 2$
👀 Show answer
$6$.
a. Write a sequence with steps of constant size in which the first term is $1$ and the fifth term is $1.04$.
$\square, \square, \square, \square, \square$
b. What is the $10$th term?
👀 Show answer
a. One correct sequence is $1, 1.01, 1.02, 1.03, 1.04$.
b. The $10$th term is $1.09$.
$7$. Ollie writes a number sequence starting at $15$ and counting back in steps of $0.4$.
$15, 14.6, 14.2, 13.8, \ldots$
He says, '$1.5$ cannot be in my sequence.'
Ollie is correct. How do you know without counting back?
Discuss your answer with your partner.
👀 Show answer
$8$. Hassan counts back in steps of $\frac{2}{5}$ starting at $0$.
He counts $0, -\frac{2}{5}, -\frac{4}{5}, -1\frac{1}{5}, \ldots$
Which of these numbers could Hassan say?
$-1\frac{4}{5} \qquad -2 \qquad -3 \qquad -3\frac{3}{5} \qquad -4$
👀 Show answer
$9$. Samira counts on from $20$ in steps of $1.001$
$20 \qquad 21.001 \qquad 22.002 \qquad 23.003 \qquad \ldots$
Write the first number Samira says which is bigger than $30$.
👀 Show answer
🧠 Think like a Mathematician
The diagram shows the first five hexagonal numbers: $1, 6, 15, 28, 45, \ldots$
Tasks:
- How does the sequence continue?
- What is the next number in the sequence?
- Write these numbers as the sum of two hexagonal numbers: $12, 21, 39, 30$.
- The first one is done for you: $12 = 6 + 6$.
- Investigate which other numbers, less than $100$, can be written as the sum of two hexagonal numbers.
👀 show answer
Sequence pattern: The differences are $+5, +9, +13, +17$, so they increase by $4$ each time. The next difference is $+21$.
Next hexagonal number:$45 + 21 = 66$.
Writing the given numbers as sums of two hexagonal numbers:
- $12 = 6 + 6$ (given)
- $21 = 15 + 6$
- $39 = 28 + 6$
- $30 = 15 + 15$
Hexagonal numbers less than 100:$1, 6, 15, 28, 45, 66, 91$
Other numbers less than 100 that can be written as the sum of two hexagonal numbers (in addition to $12, 21, 30, 39$):
$2, 7, 16, 22, 29, 34, 43, 46, 51, 57, 60, 67, 72, 73, 77, 79, 87, 90, 92, 94, 97$