Solution: Figure It Out 1.4

Section 1.4: Relations among Number Sequences

Q1. Can you find a similar pictorial explanation for why adding counting numbers up and down (1 + 2 + 1 = 4) gives square numbers?

Answer: Yes. Imagine a square grid rotated 45 degrees (like a diamond).

• The top row has 1 dot.
• The next row has 2 dots.
• The middle (widest) row has $n$ dots.
• The rows then decrease back to 1. Counting the dots row by row gives the sequence $1 + 2 + \dots + n + \dots + 2 + 1$, which forms the complete square $n^2$.

Q2. What is the value of 1 + 2 + … + 99 + 100 + 99 + … + 1?

Answer: This follows the “Up and Down” pattern which equals the square of the middle (peak) number. Here, the peak is 100. $\text{Sum} = 100 \times 100 = \mathbf{10,000}$

Q3. Which sequence do you get when you start to add the All 1’s sequence up?

Answer:

• $1$
• $1 + 1 = 2$
• $1 + 1 + 1 = 3$
• You get the Counting Numbers ($1, 2, 3, 4\dots$).

Q4. Which sequence do you get when you start to add the Counting numbers up?

Answer:

• $1$
• $1 + 2 = 3$
• $1 + 2 + 3 = 6$
• $1 + 2 + 3 + 4 = 10$
• You get the Triangular Numbers ($1, 3, 6, 10\dots$).

Q5. What happens when you add up pairs of consecutive triangular numbers? (e.g., 1+3, 3+6…)

Answer:

• $1 + 3 = 4$
• $3 + 6 = 9$
• $6 + 10 = 16$
• $10 + 15 = 25$
• You get the Square Numbers ($4, 9, 16, 25\dots$).
• Why? Two triangles can be fitted together to form a square.

Q8. What happens when you start to add up hexagonal numbers (1, 1+7, 1+7+19…)?

Answer:

• $1 = 1 = 1^3$
• $1 + 7 = 8 = 2^3$
• $1 + 7 + 19 = 27 = 3^3$
• $1 + 7 + 19 + 37 = 64 = 4^3$
• You get the Cube Numbers ($1, 8, 27, 64\dots$).