Mathematical Patterns for Coding

🎯 1. Learning Objectives

After completing this topic, you should be able to:

Recognize arithmetic, geometric and successive-difference patterns.
Use square, cube, triangular and Fibonacci patterns.
Connect a sequence term with its index and derive a formula.
Use modular arithmetic to identify repeating cycles.
Translate a mathematical pattern into loops and array logic.
Estimate loop executions before writing or analysing code.

🧩 2. What Is a Mathematical Pattern?

A mathematical pattern is a predictable relationship between values, positions or operations. In coding, identifying that relationship often replaces repeated trial-and-error with a short formula or algorithm.

Value Pattern

Each term is produced from the previous term.

3, 7, 11, 15, ... (+4)

Position Pattern

The value is calculated directly from its position n.

1, 4, 9, 16, ... a_n = n²

Repeating Pattern

A fixed cycle repeats after a known number of steps.

A, B, C, A, B, C, ... index mod 3

Structural Pattern

Rows, columns or neighbouring values follow a rule.

Pascal: value = upper-left + upper-right

➕ 3. Arithmetic and Geometric Patterns

Arithmetic Pattern

A constant difference d is added each time.

a_n = a + (n − 1)d

Example: 5, 8, 11, 14, ... has d = 3.

Geometric Pattern

Every term is multiplied by a constant ratio r.

a_n = ar^{n − 1}

Example: 2, 6, 18, 54, ... has r = 3.

Coding connection: Arithmetic patterns appear in index movement, while geometric patterns appear in repeated doubling, binary growth and divide-and-conquer analysis.

🔢 4. Square, Cube and Triangular Patterns

Position n	Square n²	Cube n³	Triangular n(n+1)/2
1	1	1	1
2	4	8	3
3	9	27	6
4	16	64	10
5	25	125	15

Sum of Odd Numbers

The first n odd numbers always add to n².

1 + 3 + 5 + ... + (2n − 1) = n²

Growing Loop Count

Executing 1, then 2, then 3 operations creates a triangular total.

1 + 2 + ... + n = n(n + 1)/2

📈 5. Successive-Difference Patterns

When the first difference is not constant, calculate differences again. A constant second difference usually indicates a quadratic pattern.

Sequence: 4, 7, 12, 19, 28, ...
First differences: 3, 5, 7, 9, ...
Next difference: 11
Next term: 28 + 11 = 39

🌱 6. Fibonacci and Recurrence

A recurrence defines a value using earlier values. The Fibonacci pattern uses the previous two terms.

F_n = F_n−1 + F_n−2, F₀ = 0, F₁ = 1

first = 0
second = 1
repeat n times:
    print first
    next = first + second
    first = second
    second = next

Important: A recurrence must have one or more starting values. Without a base value, the sequence cannot begin.

🔺 7. Pascal’s Triangle

Pascal’s Triangle begins and ends each row with 1. Every interior value is the sum of the two values directly above it.

1
1  1
1  2  1
1  3  3  1
1  4  6  4  1

The sum of row n is 2ⁿ when counting the top as row 0.
Row n contains n + 1 values.
The values represent binomial coefficients.
It is generated naturally with nested loops or dynamic programming.

🔁 8. Repeating Cycles and Modular Patterns

Alternating Values

Use index % 2 to alternate between two states.

even index → A, odd index → B

Fixed-Length Cycle

Use index % k for a cycle containing k states.

0, 1, 2, 0, 1, 2, ... → index % 3

Wrapping an Array

Move to the next position and return to zero after the last item.

next = (current + 1) % size

Checkerboard Pattern

Rows and columns alternate based on the parity of their sum.

(row + column) % 2

INTERACTIVE VISUALIZER

🧪 9. Coding Pattern Explorer

Choose a pattern and its values. The sequence and explanation appear only after you click Generate Pattern.

Pattern type

First term

Common difference

Number of terms

Select a pattern or example, then click Generate Pattern.

💻 10. Patterns in Loops and Arrays

Single Loop

A loop from 1 to n usually performs n iterations.

1 + 1 + ... + 1 = n

Growing Nested Loop

If the inner loop runs i times, the total is triangular.

1 + 2 + ... + n = n(n+1)/2

Halving Pattern

Repeatedly halving the input produces logarithmic growth.

n, n/2, n/4, ... → about log₂n steps

Pair Pattern

The number of unordered pairs from n items is triangular.

n(n−1)/2 pairs

🧮 11. Solved Problems

Problem 1: Find the next term

4, 7, 12, 19, 28, ...

Solution: Differences are 3, 5, 7 and 9. The next difference is 11, so the next term is 28 + 11 = 39.

Problem 2: Count loop executions

The inner loop runs 1, 2, 3, 4 and 5 times.

Solution: Total = 1 + 2 + 3 + 4 + 5 = 5×6/2 = 15.

Problem 3: Identify a cycle

A status sequence repeats R, G, B. Find the state at index 14.

Solution: 14 % 3 = 2. Index 2 in [R, G, B] is B.

Problem 4: Continue a recurrence

a₁ = 1 and a_n = 2a_n−1 + 1. Find a₅.

Solution: The terms are 1, 3, 7, 15 and 31.

✍️ 12. Practice Problems

Solve each problem yourself, then open its solution to verify your method.

1. Find the next term: 6, 10, 14, 18, ...

2. Find the next term: 2, 6, 12, 20, 30, ...

3. How many times does an inner loop execute if it runs i times for i = 1 to 8?

4. A cycle is [0, 1, 2, 3]. Which value appears at index 19?

5. Find the next Fibonacci term: 3, 5, 8, 13, 21, ...

📝 13. Quick Revision

Constant first difference → arithmetic pattern.
Constant ratio → geometric pattern.
Constant second difference often → quadratic pattern.
1 + 2 + ... + n = n(n+1)/2.
The first n odd numbers add to n².
Use index % k to model a repeating cycle of length k.
Always identify starting values before applying a recurrence.
Pattern recognition can replace repeated computation with a direct formula.

INTERACTIVE PRACTICE

🔺 Mathematical Patterns Practice

Complete 20 questions from easy to hard. Your score, every option, correct answers and explanations appear after final submission.

Start 20-Question Test →

🔺 Mathematical Patterns for Coding