Patterns & Sequences • Topic 3 of 3

Geometric Progressions (GP)

What are Visual Patterns?

Visual patterns are sequences of shapes, figures, or designs that follow a predictable rule. By analyzing how the pattern changes from one figure to the next, we can predict future figures and find mathematical relationships.

Why Study Visual Patterns?

Helps develop spatial reasoning and logical thinking
Connects geometry with algebra (finding formulas for patterns)
Appears in art, architecture, nature, and design
Builds foundation for understanding functions and sequences

Common Types of Visual Patterns:

Type	Description	Example
Growing Patterns	Figures increase in size or number of elements	Triangle numbers, square numbers
Repeating Patterns	Same block repeats (tessellations, borders)	ABAB pattern, striped designs
Rotational Patterns	Figures rotate by a fixed angle each step	Pinwheel designs
Symmetrical Patterns	Figures have reflection symmetry	Butterfly wings, snowflakes

Connecting Visual Patterns to Number Patterns:

Many visual patterns generate famous number sequences:

Visual Pattern	Figure 1	Figure 2	Figure 3	Number Sequence
Square numbers (dots in a square)	1 dot	4 dots	9 dots	1, 4, 9, 16, 25, ...
Triangular numbers (dots in triangle)	1 dot	3 dots	6 dots	1, 3, 6, 10, 15, ...
Rectangular numbers	2 dots	6 dots	12 dots	2, 6, 12, 20, 30, ...

Formula for Common Visual Patterns:

Square numbers: $n^2$
Triangular numbers: $\frac{n(n+1)}{2}$
Rectangular numbers: $n(n+1)$

Solved Examples

Worked Example

Example 1: Draw the next figure in the pattern: Figure 1: ●, Figure 2: ●●, Figure 3: ●●● (arranged in a row). How many dots will be in Figure 10?

Solution- Step 1: Observe the pattern: Each figure adds 1 dot to the row - Step 2: Figure 1 has 1 dot, Figure 2 has 2 dots, Figure 3 has 3 dots - Step 3: Figure 4 would have 4 dots: ●●●● - Step 4: The pattern follows $a_n = n$ - Step 5: For Figure 10, number of dots = 10

Worked Example

Example 2: A pattern of dots forms a triangle: Figure 1 has 1 dot, Figure 2 has 3 dots, Figure 3 has 6 dots, Figure 4 has 10 dots. Find the formula and number of dots in Figure 8.

Solution- Step 1: This is the triangular numbers pattern - Step 2: Sequence: 1, 3, 6, 10, 15, ... - Step 3: Differences: +2, +3, +4, +5 (not constant, so not arithmetic) - Step 4: Formula for triangular numbers: $T_n = \frac{n(n+1)}{2}$ - Step 5: Verify: $n=1$: $1×2÷2=1$; $n=2$: $2×3÷2=3$; $n=3$: $3×4÷2=6$ - Step 6: For $n=8$: $T_8 = \frac{8 \times 9}{2} = \frac{72}{2} = 36$

Worked Example

Visual description: - Figure 1: 2×2 square = 4 tiles - Figure 2: 3×3 square = 9 tiles - Figure 3: 4×4 square = 16 tiles

Solution- Step 1: Recognize the pattern: Figure n is an $(n+1) \times (n+1)$ square - Step 2: Number of tiles = (side length)² = $(n+1)^2$ - Step 3: Check: $n=1$: $(2)^2 = 4$; $n=2$: $(3)^2 = 9$; $n=3$: $(4)^2 = 16$ ✓ - Step 4: For Figure 12: $n=12$, tiles = $(12+1)^2 = 13^2 = 169$ - Step 5: Formula: $a_n = (n+1)^2$ or $a_n = n^2 + 2n + 1$

Key Points

Visual patterns use shapes/figures that follow a predictable rule
Analyze patterns by counting elements (dots, tiles, matchsticks) in each figure
Convert visual patterns to number patterns to find formulas
Common patterns: square numbers, triangular numbers, arithmetic growth
Formula for triangular numbers: $\frac{n(n+1)}{2}$
Formula for square numbers: $n^2$
Always create a table (Figure # → Count) to identify the relationship
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Quick Check — 4 MCQs

Tap an option to check your answer0 / 4

Q1.In a GP, each term is multiplied by a constant:

Explanation: Common ratio $r$.

Q2.The $n$-th term of a GP is:

Explanation: $a\,r^{n-1}$.

Q3.For $2, 6, 18$, the common ratio is:

Explanation: $r=3$.

Q4.For $3, 6, 12, 24$, $r=$

Explanation: $r=2$.

Practice more MCQs → 📝 Assignment · 35 marks