Counting and Sequences
Section: Number | Syllabus: Cambridge Primary Mathematics (0845)
Counting in Constant Steps
- A constant step means adding or subtracting the same amount each time.
- You can count in steps of:
- whole numbers, such as 4 or 7
- fractions, such as 1/2 or 3/4
- decimals, such as 0.2 or 1.5
- When counting back, you can continue past zero into negative numbers.
The constant step is +3, so 3 is added each time.
To find the step in a sequence, compare two neighbouring terms.
- 0 → 3 means +3
- 3 → 6 means +3
- 6 → 9 means +3
Because the same amount is added each time, the sequence has a constant step of 3.
Worked Example: Counting Back Through Zero
- Question: Count back in steps of 7, starting at 20. Write the first 5 terms.
The constant step is -7, so 7 is subtracted each time.
Counting in Fraction and Decimal Steps
- The same idea works when the step is a fraction or decimal.
- For example, counting on by 1/6 means add 1/6 each time:
- Counting on by 0.3 means add 0.3 each time:
The constant step is 1/6, so 1/6 is added each time.
The constant step is 0.3, so 0.3 is added each time.
Common Mistakes
MistakeStopping a counting-back sequence when it reaches zero
FixKeep using the same constant step. If the next number is below zero, write it as a negative number.
MistakeChanging the step part-way through a sequence
FixA constant step must stay the same from one term to the next.
Using Letters for Quantities
- A letter can stand for a quantity.
- We can use a letter when the quantity is unknown or when it can vary.
- The letter does not have a special value. Its value depends on the situation.
Writing Expressions with Letters
- "A number plus 5" can be written as n + 5.
- "A number minus 8" can be written as n − 8.
- "8 less than a number" can also be written as n − 8.
If n = 10, then:
- n + 5 = 10 + 5 = 15
- n − 8 = 10 − 8 = 2
Why Use a Letter?
Suppose a number changes:
- If the number is 4, then the number plus 3 is 7.
- If the number is 10, then the number plus 3 is 13.
- If the number is 25, then the number plus 3 is 28.
Instead of writing a different calculation each time, we can use n + 3 to describe the rule for any value of n.
Worked Example: Using a Letter
-
Question: A number is represented by n. Write an expression for "6 more than the number".
- Step 1: The number is represented by n.
- Step 2: "6 more than" means add 6.
- Answer: n + 6
Letters make it possible to describe an addition or subtraction calculation even when the quantity is not fixed.
Common Mistakes
MistakeWriting 6n for "6 more than n"
Fix6n means 6 multiplied by n. "6 more than n" means n + 6.
MistakeThinking a letter always represents an unknown number
FixA letter can represent a quantity that is unknown or a quantity that can change.
Position-to-Term Rules
- A sequence can have a term-to-term rule that tells us how to get from one term to the next.
- A position-to-term rule tells us how to find a term directly from its position.
- Repeated addition of the same amount can be represented using multiplication.
From Repeated Addition to Multiplication
Look at the sequence:
3, 6, 9, 12, 15, ...
- The sequence increases by 3 each time.
- 1st term = 3 × 1
- 2nd term = 3 × 2
- 3rd term = 3 × 3
- 4th term = 3 × 4
So the position-to-term rule is:
3 × position
When the Sequence Does Not Start at the Step Size
Consider the sequence:
4, 7, 10, 13, 16, ...
- The sequence increases by 3, so start with 3 × position.
- 3 × 1 = 3, but the first term is 4.
- We need to add 1 to every result.
- Therefore, the position-to-term rule is 3n + 1, where n represents the position.
| Position (n) | 3n + 1 | Term |
|---|---|---|
| 1 | 3(1) + 1 | 4 |
| 2 | 3(2) + 1 | 7 |
| 3 | 3(3) + 1 | 10 |
| 4 | 3(4) + 1 | 13 |
Worked Example: Finding the 10th Term
-
Question: The sequence is 4, 7, 10, 13, ... Find the 10th term.
- Step 1: The sequence increases by 3, so the rule starts with 3n.
- Step 2: 3n gives 3 for position 1, but the first term is 4. Add 1.
- Step 3: The position-to-term rule is 3n + 1.
- Step 4: Substitute n = 10: 3(10) + 1 = 30 + 1 = 31.
- Answer: The 10th term is 31.
Term-to-Term or Position-to-Term?
- Term-to-term: "Add 3 each time" moves from one term to the next.
- Position-to-term: "3n + 1" lets you find a term directly from its position.
A position-to-term rule is especially useful when you want a term far along in the sequence, such as the 100th term.
Common Mistakes
MistakeGiving the term-to-term rule when asked for the position-to-term rule
Fix"Add 3" tells you how to move between terms. A rule such as 3n + 1 connects the position directly to the term.
MistakeForgetting the extra number in a rule such as 3n + 1
FixCheck the rule using the first term. If 3(1) = 3 but the first term is 4, the rule needs +1.
Sequences from Square Numbers
- A square number is the result of multiplying a whole number by itself.
- For example:
- 1 × 1 = 1, so 1 is a square number.
- 2 × 2 = 4, so 4 is a square number.
- 3 × 3 = 9, so 9 is a square number.
- 4 × 4 = 16, so 16 is a square number.
- We can write these using square notation:
- 1² = 1
- 2² = 4
- 3² = 9
- 4² = 16
The position tells us the side length of the square, so the number of dots is the position multiplied by itself.
Using Square Numbers in a Sequence
Sometimes a sequence is made by taking a square number and then applying another rule.
For example:
nth term = n² + 2
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