**Key Number Sequences to Remember:**
• **All 1's:** 1, 1, 1, 1, 1... (same number repeats)
• **Counting numbers:** 1, 2, 3, 4, 5... (add 1 each time)
• **Odd numbers:** 1, 3, 5, 7, 9... (add 2 each time)
• **Even numbers:** 2, 4, 6, 8, 10... (add 2 each time)
• **Triangular numbers:** 1, 3, 6, 10, 15, 21... (arrange as triangle dots)
• **Square numbers:** 1, 4, 9, 16, 25, 36... (1², 2², 3², 4², etc.)
• **Cube numbers:** 1, 8, 27, 64, 125... (1³, 2³, 3³, etc.)
• **Powers of 2:** 1, 2, 4, 8, 16, 32, 64... (each = previous × 2)
• **Powers of 3:** 1, 3, 9, 27, 81... (each = previous × 3)
**Important Pattern: Sum of Odd Numbers = Squares**
1 = 1² | 1+3 = 2² | 1+3+5 = 3² | 1+3+5+7 = 4²
**Important Pattern: Sum Up-and-Down = Squares**
1 = 1² | 1+2+1 = 2² | 1+2+3+2+1 = 3² | 1+2+3+4+3+2+1 = 4²
**Diagrams to Remember:** Draw square grids with dots to visualise square numbers, arrange dots in triangles to show triangular numbers.
**Don't Confuse:** Odd numbers (1,3,5,7...) ≠ Square numbers (1,4,9,16...); both odd and even numbers have their own patterns.
Q1. Which of the following is a square number?
Answer: B — 25 is a square number because 5 × 5 = 25, whereas 15, 35, and 45 are not perfect squares.
Q2. In the sequence of odd numbers 1, 3, 5, 7, 9, ..., what is the 10th odd number?
Answer: B — The 10th odd number is found by the pattern: odd number = 2n – 1, where n = 10, so 2(10) – 1 = 19.
Q3. What pattern do you observe when adding the first few odd numbers: 1 = 1, 1 + 3 = 4, 1 + 3 + 5 = 9?
Answer: C — The sum of consecutive odd numbers starting from 1 always produces square numbers: 1, 4, 9, 16, 25, etc.
Q4. Which number can be arranged both as a perfect square AND as a triangular number?
Answer: C — 36 is special because 36 = 6² (a square) and 36 = 1+2+3+4+5+6+7+8 (a triangular number).
Q5. In the sequence 2, 4, 6, 8, 10, ..., this sequence is called _____.
Answer: C — This sequence contains only even numbers, which are numbers divisible by 2.
Q6. What is the next number in the sequence of powers of 2: 1, 2, 4, 8, 16, 32, ___?
Answer: C — In powers of 2, each number is multiplied by 2 to get the next, so 32 × 2 = 64.
Q7. A number sequence that can be arranged as dots in a triangular pattern is called a _____ number.
Answer: B — Triangular numbers (1, 3, 6, 10, 15...) get their name because the dots can be arranged in a triangle shape.
Q8. Looking at the pattern 1, 3, 6, 10, 15, 21, ..., which term comes next?
Answer: C — These are triangular numbers; each term adds the next counting number: 21 + 7 = 28.
Q9. In the sequence of cube numbers 1, 8, 27, 64, ..., the number 27 is 3 cubed. What does '3 cubed' mean?
Answer: C — 3 cubed (3³) means 3 × 3 × 3 = 27, where a number is multiplied by itself three times.
Q10. Why do mathematicians study number patterns and sequences?
Answer: B — Mathematicians study patterns to both discover what patterns exist and understand why they exist, which has real-world applications in science, nature, and technology.
What is a number sequence?
A number sequence is an ordered list of numbers that follow a specific pattern or rule.
What are odd numbers?
Odd numbers are numbers that cannot be divided equally by 2, like 1, 3, 5, 7, 9, 11, 13.
What are even numbers?
Even numbers are numbers that can be divided equally by 2, like 2, 4, 6, 8, 10, 12, 14.
What are triangular numbers?
Triangular numbers are 1, 3, 6, 10, 15, 21, 28... — numbers that can be arranged in a triangular shape with dots.
What are square numbers?
Square numbers are 1, 4, 9, 16, 25, 36... — numbers that result from multiplying a number by itself (1×1, 2×2, 3×3, etc.).
What are cube numbers?
Cube numbers are 1, 8, 27, 64, 125... — numbers that result from multiplying a number by itself three times (1×1×1, 2×2×2, etc.).
Why is 36 special in number patterns?
36 is special because it is both a square number (6×6) and a triangular number (1+2+3+4+5+6+7+8), so it can be arranged as both a square and a triangle.
What pattern do you get when you add odd numbers: 1 + 3 + 5 + 7 + 9 + 11?
When you add consecutive odd numbers starting from 1, you always get square numbers (1=1, 1+3=4, 1+3+5=9, 1+3+5+7=16, etc.).
What is the rule for the sequence 1, 2, 4, 8, 16, 32, 64?
This is the sequence of powers of 2, where each number is made by multiplying the previous number by 2.
Why do mathematicians study number patterns?
Mathematicians study patterns both to find what patterns exist and to understand why they exist, which helps explain many things in nature and science.
What is a number sequence? Give one example. [1 mark]
A number sequence is an ordered list of numbers following a pattern; examples include odd numbers (1, 3, 5, 7...), even numbers (2, 4, 6, 8...), or counting numbers (1, 2, 3, 4...).
Write the first 6 terms of the sequence of square numbers. How is each number formed? [2 marks]
Square numbers are 1, 4, 9, 16, 25, 36; each is formed by multiplying a counting number by itself (1×1, 2×2, 3×3, 4×4, 5×5, 6×6).
Add the first 5 odd numbers (1 + 3 + 5 + 7 + 9) and observe the pattern. Explain with a reason why this pattern occurs. Draw a diagram to show your explanation. [3 marks]
1 + 3 + 5 + 7 + 9 = 25 (which is 5²); draw a square grid and partition the dots into layers of odd numbers to show why the sum gives square numbers.
The sequence 1, 3, 6, 10, 15, 21, ... is called triangular numbers. (a) Why are they called triangular numbers? (b) Draw the first four triangular numbers using dots arranged in triangle shapes. (c) What is the next triangular number after 21? Explain the rule used to find it. [5 marks]
Triangular numbers are called that because dots can be arranged in triangle shapes; draw dots in triangular patterns (1 dot, 1+2 dots, 1+2+3 dots, etc.); the next term is found by adding the next counting number (21 + 7 = 28).
True or False: Every square number is also an odd number. Give a reason for your answer. [2 marks]
False — square numbers include 4, 16, 36, which are even; only odd-positioned squares like 1, 9, 25 are odd.
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