How many little triangles are there in each shape of the sequence of Stacked Triangles?

How many little triangles are there in each shape of the sequence of Stacked Triangles? Class 6

Question 1.

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?) Class 6

Solution:

Sequence of square numbers 1, 4, 9, 16, ...

Question 2.

To get from one shape to the next shape in the Koch Snowflake sequence, one replaces each line segment ‘—’ by a ‘speed bump’ __/\__. As one does this more and more times, the changes become tinier and tinier with very, very small line segments. How many total line segments are there in each shape of the Koch Snowflake? What is the corresponding number sequence? (The answer is 3, 12, 48, 192, ..., i.e., 3 times Powers of 4; this sequence is not shown in Table 1) Class 6

Solution:

Sequence 3, 12, 48, 192, ...

That is, it is 3 × 1, 3 × 4, 3 × 16, 3 × 64, ...

In other words, it is the sequence of 3 × 4⁰, 3 × 4¹, 3 × 4², 3 × 4³, ... or the sequence of 3 times the powers of 4.