MIP1502 Assignment 3 due 18 July 2025

MIP1502
ASSIGNMENT 3 2025

UNIQUE NO. 721003
DUE DATE: 18 JULY 2025

,MIP1502 Assignment 3

QUESTION 1: PATTERNS AND ALGEBRA

1.1.1.1: First Differences Between Consecutive Terms

 Small triangles: 1, 4, 9, 16, 25, 36
Differences: 4 - 1 = 3, 9 - 4 = 5, 16 - 9 = 7, 25 - 16 = 9, 36 - 25 = 11
First differences: 3, 5, 7, 9, 11
 Black triangles: 1, 3, 5, 7, 9, 11
Differences: 3 - 1 = 2, 5 - 3 = 2, 7 - 5 = 2, 9 - 7 = 2, 11 - 9 = 2
First differences: 2, 2, 2, 2, 2
 Grey triangles: 0, 1, 3, 6, 10
Differences: 1 - 0 = 1, 3 - 1 = 2, 6 - 3 = 3, 10 - 6 = 4
First differences: 1, 2, 3, 4
 White triangles: 0, 0, 1, 3, 6, 10
Differences: 0 - 0 = 0, 1 - 0 = 1, 3 - 1 = 2, 6 - 3 = 3, 10 - 6 = 4
First differences: 0, 1, 2, 3, 4

1.1.1.2: Classifying Patterns

 Small triangles: Quadratic (because the first differences are not constant, but
the second differences are).
 Black triangles: Linear (because the first differences are constant).
 Grey triangles: Quadratic
 White triangles: Quadratic

1.1.1.3: Justification for Classification

 The black triangle pattern increases by the same amount each time (2), which is
a constant first difference. This tells us it's a linear pattern, because it grows
steadily.

,  The small triangle pattern doesn’t increase by the same amount. Instead, the
difference between each step increases (3, 5, 7, etc.). But if you look at the
second difference (2, 2, 2), that’s constant. This means it's a quadratic pattern,
because its growth gets faster and faster in a predictable way.




1.1.2.1: Describing How the Grey and White Triangle Patterns Grow

 Grey triangles: In each diagram, you add more grey triangles than before. For
example, from diagram 2 to 3, you go from 1 to 3 grey triangles (you added 2).
Then you add 3, then 4. So it's like you're adding more and more each time.
 White triangles: This one starts even slower. Nothing in the first two diagrams,
then you add 1 white triangle, then 2, then 3, then 4, etc. So again, it's
increasing, but more slowly than the grey ones.

1.1.2.2: Flow Diagrams

Let’s create simple diagrams. The input is the diagram number (n). We use the formula
for triangular numbers: T(n) = n(n - 1)/2 for grey, and T(n) = (n - 2)(n - 1)/2 for white
(from n ≥ 3).

Grey Triangle Flow Diagram

Input n → Multiply by (n - 1) → Divide by 2 → Output: Number of Grey Triangles

Formula: T(n) = n(n - 1)/2

White Triangle Flow Diagram

Input n → Subtract 2 → Multiply by (n - 1) → Divide by 2 → Output: Number of White
Triangles

Formula: T(n) = (n - 2)(n - 1)/2