MATH HELP. Triangles and line segments?

Question

Craig constructs the designs above from equal line segments. The design in Pattern 1 is made up of three line segments. Pattern 2 is made up of nine line segments. Pattern 3 is made up of thirty line segments, and so on.
How many line segments are needed to make Pattern 8?
How many line segments are needed to make pattern 16?

Determine a function for finding the number of line segments needed to make pattern for any number n. Justify why your function works.

Answer 1

From your description, the patterns must be composed of a pyramid of triangles. I'm imagining that they look like the diagram below.



The number of triangles in the base row seems to be doubling (1, 2, 4, 8, etc.) so that would be 2^(k-1) triangles. The total number of colored triangles (facing up) that are stacked up would be the nth triangular numbers (1, 3, 6, 10, 15, 21, etc.) which are figured using this formula:

Tn = n(n+1) / 2



And finally we need to multiply everything by 3 to account for the 3 sides of each triangle.



PART A:



Pattern 8 will have 2^7 triangles in the base row. We want the 128th triangular number:

T128 = 128 * 129/2

= 8,256 triangles

Multiply that by 3 sides



Answer:

Pattern 8 = 24,768 segments



PART B:



Pattern 16 will have 2^15 triangles in the base row. We want the 32768th triangular number:

T32768 = 32768 * 32769/2

= 536,887,296 triangles

Multiply that by 3 sides



Answer:

Pattern 16 = 1,610,661,888 segments



PART C:

Putting everything together we have:

f(n) = 3/2 * [2^(k-1) * (2^(k-1) + 1) ]



As a double-check, here are the segments for the first few patterns:

1, 3

2, 9

3, 30

4, 108

5, 408

6, 1584

7, 6240

8, 24768

9, 98688

10, 393984

11, 1574400

12, 6294528

13, 25171968

14, 100675584

15, 402677760

16, 1610661888

17, 6442549248

18, 25770000384

Answer 2

Equal Line Segments

Answer 3

Doubt anyone can help you without a picture of the designs.