Sequences and Patterns

What is a Sequence?

A sequence is an ordered list of numbers following a rule.

Terms: The individual numbers in a sequence

• First term (a₁), second term (a₂), third term (a₃), etc.

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

• a₁ = 3, a₂ = 7, a₃ = 11

Arithmetic Sequences

Arithmetic sequence: Add the same number each time

Common difference (d): The amount added each step

Formula: d = a₂ − a₁

Example 1: Identify Arithmetic Sequence

Sequence: 5, 8, 11, 14, 17...

Check:

• 8 − 5 = 3
• 11 − 8 = 3
• 14 − 11 = 3

Common difference: d = 3

Yes, it's arithmetic!

Example 2: Find Next Terms

Sequence: 10, 15, 20, 25, ...

Common difference: 15 − 10 = 5

Next terms:

• 25 + 5 = 30
• 30 + 5 = 35

Answer: 30, 35

Example 3: Decreasing Sequence

Sequence: 100, 94, 88, 82, ...

Common difference: 94 − 100 = −6

Next term: 82 + (−6) = 76

Answer: 76

Arithmetic Sequence Formula

Find any term without listing them all!

Formula: aₙ = a₁ + (n − 1)d

Where:

• aₙ = nth term
• a₁ = first term
• n = term number
• d = common difference

Example 1: Find 20th Term

Sequence: 3, 7, 11, 15... (d = 4, a₁ = 3)

Find a₂₀:

a₂₀ = 3 + (20 − 1)(4)
a₂₀ = 3 + 19(4)
a₂₀ = 3 + 76
a₂₀ = 79

Answer: The 20th term is 79

Example 2: Find 50th Term

Sequence: 2, 5, 8, 11... (d = 3, a₁ = 2)

Find a₅₀:

a₅₀ = 2 + (50 − 1)(3)
a₅₀ = 2 + 49(3)
a₅₀ = 2 + 147
a₅₀ = 149

Answer: 149

Geometric Sequences

Geometric sequence: Multiply by the same number each time

Common ratio (r): The number you multiply by

Formula: r = a₂ / a₁

Example 1: Identify Geometric Sequence

Sequence: 2, 6, 18, 54...

Check:

• 6 ÷ 2 = 3
• 18 ÷ 6 = 3
• 54 ÷ 18 = 3

Common ratio: r = 3

Yes, it's geometric!

Example 2: Find Next Terms

Sequence: 5, 10, 20, 40, ...

Common ratio: 10 ÷ 5 = 2

Next terms:

• 40 × 2 = 80
• 80 × 2 = 160

Answer: 80, 160

Example 3: Fractional Ratio

Sequence: 64, 32, 16, 8, ...

Common ratio: 32 ÷ 64 = 1/2

Next term: 8 × (1/2) = 4

Answer: 4

Geometric Sequence Formula

Formula: aₙ = a₁ × r^(n−1)

Where:

• aₙ = nth term
• a₁ = first term
• n = term number
• r = common ratio

Example 1: Find 8th Term

Sequence: 3, 12, 48, 192... (r = 4, a₁ = 3)

Find a₈:

a₈ = 3 × 4^(8−1)
a₈ = 3 × 4⁷
a₈ = 3 × 16,384
a₈ = 49,152

Answer: 49,152

Example 2: Decreasing Sequence

Sequence: 1000, 100, 10, 1... (r = 0.1, a₁ = 1000)

Find a₅:

a₅ = 1000 × (0.1)⁴
a₅ = 1000 × 0.0001
a₅ = 0.1

Answer: 0.1

Arithmetic vs. Geometric

How to tell the difference:

Arithmetic: Same difference between terms

• Subtract to check

Geometric: Same ratio between terms

• Divide to check

Example: Identify Type

Sequence A: 2, 5, 8, 11...

• Differences: 3, 3, 3 → Arithmetic (d = 3)

Sequence B: 2, 6, 18, 54...

• Ratios: 3, 3, 3 → Geometric (r = 3)

Sequence C: 1, 4, 9, 16...

• Differences: 3, 5, 7 (not constant)
• Ratios: 4, 2.25, 1.78... (not constant)
• Neither! (These are perfect squares)

Finding Position of a Term

Given a term value, find its position!

Example: Which Term is 47?

Sequence: 3, 7, 11, 15... (arithmetic, d = 4)

Use formula: 47 = 3 + (n − 1)(4)

Solve:

47 = 3 + 4n − 4
47 = 4n − 1
48 = 4n
n = 12

Answer: 47 is the 12th term

Number Patterns

Other common patterns:

Square numbers: 1, 4, 9, 16, 25... (n²) Triangular numbers: 1, 3, 6, 10, 15... (add consecutive numbers) Fibonacci: 1, 1, 2, 3, 5, 8, 13... (add previous two)

Example: Triangular Numbers

Pattern: 1, 3, 6, 10, 15...

Rule: Add next integer

• 1
• 1 + 2 = 3
• 3 + 3 = 6
• 6 + 4 = 10
• 10 + 5 = 15

Next: 15 + 6 = 21

Sum of Arithmetic Sequence

Sum of first n terms: Sₙ = n(a₁ + aₙ)/2

Example: Sum First 10 Terms

Sequence: 2, 5, 8, 11... (d = 3)

Find a₁₀: 2 + (10−1)(3) = 29

Find S₁₀:

S₁₀ = 10(2 + 29)/2
S₁₀ = 10(31)/2
S₁₀ = 155

Answer: Sum = 155

Real-World Applications

Savings: Deposit $50/month

• Month 1: $50
• Month 2: $100
• Arithmetic sequence

Population growth: Doubles each year

• Year 1: 100
• Year 2: 200
• Geometric sequence

Seating: Theater rows

• Row 1: 20 seats
• Row 2: 22 seats
• Arithmetic (d = 2)

Practice