🔄Intermediate

Number Patterns & Sequences

Patterns are the heartbeat of mathematics. Learn to spot them, extend them, and use them.

What Is a Sequence?

A sequence is an ordered list of numbers that follows a rule. Each number is called a term.

2, 4, 6, 8, 10, ... (the "..." means it continues)

Arithmetic Sequences

Each term is found by adding a constant (called the common difference).

Example 1

Sequence: 3, 7, 11, 15, 19, ...

Common difference: 7-3 = 4, 11-7 = 4, 15-11 = 4. d = 4

Next term: 19 + 4 = 23

Finding the nth Term

Formula: aₙ = a₁ + (n-1)d

Where a₁ = first term, d = common difference, n = term number.

Example 2

Find the 20th term of: 5, 9, 13, 17, ...

a₁ = 5, d = 4

a₂₀ = 5 + (20-1)(4) = 5 + 76 = 81

Sum of Arithmetic Sequence

Formula: S = (n/2)(first + last)

Sum of first 100 positive integers: S = (100/2)(1 + 100) = 50 × 101 = 5,050

Geometric Sequences

Each term is found by multiplying by a constant (called the common ratio).

Example 3

Sequence: 3, 6, 12, 24, 48, ...

Common ratio: 6/3 = 2, 12/6 = 2, 24/12 = 2. r = 2

Next term: 48 × 2 = 96

Finding the nth Term

Formula: aₙ = a₁ × r^(n-1)

Example 4

Find the 8th term of: 5, 15, 45, ...

a₁ = 5, r = 3

a₈ = 5 × 3⁷ = 5 × 2,187 = 10,935

Geometric sequences grow FAST! This is exponential growth.

The Fibonacci Sequence

Each term is the sum of the two before it: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, ...

Rule: F(n) = F(n-1) + F(n-2)

The Fibonacci sequence appears everywhere in nature:

Petals on flowers (3, 5, 8, 13 are most common)
Spiral patterns in sunflowers and pinecones
Branching patterns in trees
Nautilus shell spirals

Other Common Patterns

Perfect Squares

1, 4, 9, 16, 25, 36, 49, 64, 81, 100, ...

Pattern: 1², 2², 3², 4², ... Differences: 3, 5, 7, 9 (odd numbers!)

Triangular Numbers

1, 3, 6, 10, 15, 21, 28, ...

Each is the sum of counting numbers: 1, 1+2, 1+2+3, 1+2+3+4, ...

Powers of 2

1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, ...

Fundamental in computing (bytes, memory, etc.)

How to Identify a Pattern

Step 1: Calculate differences between consecutive terms.
If differences are constant → arithmetic
Step 2: If not, calculate ratios between consecutive terms.
If ratios are constant → geometric
Step 3: If neither, look at second differences, or check for squares, cubes, or Fibonacci-like patterns.

Example 5: Identify and extend

Sequence: 2, 5, 10, 17, 26, ...

Differences: 3, 5, 7, 9 (not constant — not arithmetic)

Second differences: 2, 2, 2 (constant! — this is quadratic: n² + 1)

Next difference: 11. Next term: 26 + 11 = 37

🎯 Try It Yourself

Test your understanding with these practice problems.

1. What comes next: 3, 7, 11, 15, 19, ?

💡 Hint: Adding 4 each time (arithmetic sequence)

2. What comes next: 2, 6, 18, 54, ?

💡 Hint: Multiplying by 3 each time (geometric sequence)

3. What comes next in Fibonacci: 1, 1, 2, 3, 5, 8, 13, ?

💡 Hint: Add the two previous numbers: 8 + 13

4. Find the 10th term: 5, 8, 11, 14, ...

💡 Hint: Formula: a + (n-1)d = 5 + 9(3) = 32

5. What comes next: 1, 4, 9, 16, 25, ?

💡 Hint: These are perfect squares: 1², 2², 3², 4², 5², 6²

⚠️ Common Mistakes

✗Assuming every pattern that increases is arithmetic. It might be geometric, quadratic, or something else entirely.
✗In arithmetic sequences, using the wrong common difference — always check multiple pairs.
✗In geometric sequences, confusing the ratio with the difference. 2, 6, 18 has ratio 3, not difference 4.
✗Off-by-one errors in the nth term formula. The 10th term uses (n-1) = 9, not 10.
✗Seeing a pattern after just 2-3 numbers and assuming it continues. You need at least 3-4 terms to be confident.

🌍 Real Life Example

Saving Money with Patterns

You start a savings challenge: save $5 the first week, $10 the second, $15 the third, and so on (increase by $5 each week). This is an arithmetic sequence. By week 52, you'd save $260 that week alone! Total for the year: sum = (52/2)(5 + 260) = 26 × 265 = $6,890. Understanding the pattern lets you predict exactly how much you'll have saved at any point.

💡 Key Takeaway

Arithmetic sequences add a constant (common difference). Geometric sequences multiply by a constant (common ratio). The Fibonacci sequence adds the two previous terms. To identify a pattern: look at differences between terms first, then ratios. Patterns appear everywhere in nature, finance, and computer science.

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