Number Pattern

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Number Pattern Theory & Formulas

1. Arithmetic Sequences

Definition

A sequence where each term increases or decreases by a constant difference.

an = a₁ + (n-1)d

Where:

• an = nth term
• a₁ = first term
• d = common difference
• n = position of term

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

3 → +4 → 7 → +4 → 11 → +4 → 15

Here: a₁ = 3, d = 4

Next term: a₆ = 3 + (6-1)×4 = 3 + 20 = 23

2. Geometric Sequences

Definition

A sequence where each term is multiplied by a constant ratio.

an = a₁ × r^(n-1)

Where:

• an = nth term
• a₁ = first term
• r = common ratio
• n = position of term

Example: 2, 6, 18, 54, 162, ...

2 → ×3 → 6 → ×3 → 18 → ×3 → 54

Here: a₁ = 2, r = 3

Next term: a₆ = 2 × 3^(6-1) = 2 × 243 = 486

3. Square Number Patterns

Definition

Sequences based on perfect squares of consecutive integers.

an = (a + n - 1)²

Where:

• a = starting number
• n = position in sequence

Example: 1, 4, 9, 16, 25, ...

1² → 2² → 3² → 4² → 5²

Pattern: 1², 2², 3², 4², 5², ...

Next terms: 6² = 36, 7² = 49

4. Fibonacci-Like Patterns

Definition

Each term is the sum of the two preceding terms.

an = an-1 + an-2

Where:

• an = current term
• an-1 = previous term
• an-2 = term before previous

Example: 2, 3, 5, 8, 13, ...

2 + 3 = 5 → 3 + 5 = 8 → 5 + 8 = 13

Next terms: 8 + 13 = 21, 13 + 21 = 34

5. Decreasing Arithmetic Patterns

Definition

Arithmetic sequence with negative common difference.

an = a₁ - (n-1)|d|

Where:

• |d| = absolute value of difference
• Difference is subtracted each time

Example: 20, 17, 14, 11, 8, ...

20 → -3 → 17 → -3 → 14 → -3 → 11

Here: a₁ = 20, d = -3

Next terms: 8 - 3 = 5, 5 - 3 = 2

6. Powers of 2 (Doubling) Pattern

Definition

Each term is double the previous term.

an = a₁ × 2^(n-1)

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

3 → ×2 → 6 → ×2 → 12 → ×2 → 24

This is geometric with r = 2

Next terms: 48 × 2 = 96, 96 × 2 = 192

7. Cube Number Patterns

Definition

Sequences based on perfect cubes of consecutive integers.

an = (a + n - 1)³

Example: 1, 8, 27, 64, 125, ...

1³ → 2³ → 3³ → 4³ → 5³

Pattern: 1³, 2³, 3³, 4³, 5³, ...

Next terms: 6³ = 216, 7³ = 343

8. Alternating Patterns

Definition

Pattern that alternates between two different rules.

Odd positions: an = f₁(n)
Even positions: an = f₂(n)

Example: 2, 8, 4, 10, 6, 12, ...

2 → +6 → 8 → -4 → 4 → +6 → 10

Odd positions: 2, 4, 6, ... (add 2)

Even positions: 8, 10, 12, ... (add 2, but start at 8)

Problem-Solving Steps

1. Find the differences between consecutive terms
2. Check if differences are constant (arithmetic)
3. Check if ratios are constant (geometric)
4. Look for special patterns (squares, cubes, Fibonacci)
5. Test your pattern with given terms
6. Apply the pattern to find next terms

Tip: Always verify your answer by checking if it fits the established pattern!

Common Pattern Types Summary

• +/- constant: Arithmetic sequence
• ×/÷ constant: Geometric sequence
• Add previous two: Fibonacci-like
• Perfect squares: n² pattern
• Perfect cubes: n³ pattern
• Powers of 2: Doubling pattern
• Mixed rules: Alternating pattern