How many little triangles are there in each shape of the sequence of Stacked Triangles? Which number sequence does this give? Can you explain why? (Hint: In each shape in the sequence, how many triangles are there in each row?)
Counting little triangles in Stacked Triangles:
First shape: 1 triangle Second shape: 4 triangles (1+3) Third shape: 9 triangles (1+3+5) Fourth shape: 16 triangles (1+3+5+7) Fifth shape: 25 triangles (1+3+5+7+9)
Sequence: 1, 4, 9, 16, 25, ...
Answer: This is the Square numbers sequence.
Explanation: Each row in the stacked triangle contains an odd number of triangles. Row 1 has 1 triangle Row 2 has 3 triangles Row 3 has 5 triangles And so on...
The total number of triangles equals the sum of consecutive odd numbers: 1 = 1² 1+3 = 4 = 2² 1+3+5 = 9 = 3² 1+3+5+7 = 16 = 4²
∴ The sum of consecutive odd numbers starting from 1 always gives a square number.
Explanation
The chapter establishes in section 1.4 that adding consecutive odd numbers gives square numbers: 1=1², 1+3=4=2², 1+3+5=9=3², etc. In Stacked Triangles, each row adds an odd number of triangles (row 1 has 1, row 2 has 3, row 3 has 5, etc.). The total triangles in each shape is the sum of odd numbers, which equals a square number.
Solution Steps
Count the little triangles in each shape of the Stacked Triangles sequence.
Identify the pattern: 1, 4, 9, 16, 25, ...
Recognize this as the Square numbers sequence.
Explain why: each row adds an odd number of triangles (1, 3, 5, 7, ...).
Conclude that the sum of consecutive odd numbers starting from 1 gives square numbers.