Solution: Figure It Out 1.3

Section 1.3: Visualising Number Sequences

Q2. Why are 1, 3, 6, 10… called triangular numbers? Why are 1, 4, 9… called square numbers? Why are 1, 8, 27… called cubes?

Answer:

• They are called Triangular Numbers because that specific number of dots can be arranged to form a perfect equilateral triangle.
• They are called Square Numbers because the dots can form a perfect square grid with equal rows and columns.
• They are called Cubes because unit blocks of that quantity can form a solid 3D cube ($2\times2\times2=8$, etc.).

Q3. You will have noticed that 36 is both a triangular number and a square number!

Answer:

• As a Square: $6 \times 6 = 36$.
• As a Triangle: Sum of $1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 = 36$.

Q4. What would you call the following sequence of numbers: 1, 7, 19, 37…?

Answer: These are called Hexagonal Numbers.

• Next Number: The pattern of differences is multiples of 6 ($+6, +12, +18$). The next difference is $+24$.
• $37 + 24 = \mathbf{61}$.

Q5. Can you think of pictorial ways to visualise the sequence of Powers of 2? Powers of 3?

Answer:

• Powers of 2 ($1, 2, 4, 8$): Can be visualized as a line branching into 2, then those branching into 2 (Binary Tree), or identifying dimensions (Point, Line, Square, Cube corners).
• Powers of 3 ($1, 3, 9, 27$): Can be visualized as a triangle splitting into 3 smaller triangles, or a tree where every branch splits into three.