00-13 — Number Sequences and Patterns

Phase: 0 — Arithmetic & Number Foundations Subject: 00-13 Prerequisites: 00-07 — Ratios, Rates, and Proportions Next subject: 00-14 — Factors, Multiples, and Primes (Deep Dive)

Learning Objectives

By the end of this subject, you will be able to:

1. Identify whether a sequence is arithmetic, geometric, or neither, and determine its common difference or common ratio
2. Find the nth term (general term) of an arithmetic or geometric sequence using the standard formulas
3. Calculate any specific term of a sequence given its position, or find the position given the term value
4. Recognise and extend famous sequences including Fibonacci, triangular numbers, and square numbers
5. Analyse visual and numerical patterns to derive the rule governing a sequence

Core Content

1. What Is a Sequence?

A sequence is an ordered list of numbers following a specific rule. Each number in the sequence is called a term.

Notation: - The first term is T₁ (or a₁), second is T₂, third is T₃, etc. - The nth term (general term) is Tₙ (or aₙ) - A sequence with n terms: T₁, T₂, T₃, ..., Tₙ

Examples of sequences: - 2, 4, 6, 8, 10, ... (even numbers — rule: add 2 each time) - 3, 7, 11, 15, 19, ... (rule: add 4 each time) - 1, 3, 9, 27, 81, ... (rule: multiply by 3 each time) - 1, 1, 2, 3, 5, 8, 13, ... (Fibonacci — each term is sum of previous two)

⚠️ THIS IS CRITICAL — Sequences are the gateway to functions and algebra. Understanding how to find the nth term is essentially understanding how to write a function f(n) that gives the nth value. This concept appears in algebra, calculus (series), computer science (algorithms), and probability.

2. Arithmetic Sequences

An arithmetic sequence (also called an arithmetic progression or AP) is a sequence where the difference between consecutive terms is constant. This constant is the common difference d.

Tₙ₊₁ − Tₙ = d (constant for all n)

Identifying an arithmetic sequence: Subtract each term from the next. If the difference is always the same, it's arithmetic.

Example 1: 5, 9, 13, 17, 21, ...

$9 − 5 = 4, 13 − 9 = 4, 17 − 13 = 4, 21 − 17 = 4
Constant difference d = 4. This is arithmetic.
$

Example 2: 20, 15, 10, 5, 0, −5, ...

$15 − 20 = −5, 10 − 15 = −5, 5 − 10 = −5
Constant difference d = −5. This is arithmetic (decreasing).
$

Example 3: 2, 4, 8, 16, 32, ...

$4 − 2 = 2, 8 − 4 = 4 — not constant. NOT arithmetic.
$

3. The nth Term of an Arithmetic Sequence

For an arithmetic sequence with first term a and common difference d:

Tₙ = a + (n − 1)d

Where does this formula come from?

T₁ = a
T₂ = a + d
T₃ = a + 2d
T₄ = a + 3d
...
Tₙ = a + (n − 1)d     ← n−1 jumps of size d from the first term

Example 1: Find the 20th term of 3, 7, 11, 15, ...

$a = 3, d = 4
T₂₀ = 3 + (20 − 1) × 4 = 3 + 19 × 4 = 3 + 76 = 79
$

Example 2: Find the nth term of 10, 6, 2, −2, ...

$a = 10, d = −4
Tₙ = 10 + (n − 1)(−4) = 10 − 4(n − 1) = 10 − 4n + 4 = 14 − 4n
$

Example 3: The 5th term of an arithmetic sequence is 23 and the 12th term is 65. Find a and d.

$T₅ = a + 4d = 23
T₁₂ = a + 11d = 65

Subtract: (a + 11d) − (a + 4d) = 65 − 23
         7d = 42
         d = 6

Then: a + 4(6) = 23 → a + 24 = 23 → a = −1

The sequence is: −1, 5, 11, 17, 23, 29, 35, 41, 47, 53, 59, 65, ...
Tₙ = −1 + (n − 1) × 6 = 6n − 7
$

4. Geometric Sequences

A geometric sequence (also called a geometric progression or GP) is a sequence where the ratio of consecutive terms is constant. This constant is the common ratio r.

Tₙ₊₁ / Tₙ = r (constant for all n)

Identifying a geometric sequence: Divide each term by the previous term. If the ratio is always the same, it's geometric.

Example 1: 3, 6, 12, 24, 48, ...

$6 ÷ 3 = 2, 12 ÷ 6 = 2, 24 ÷ 12 = 2, 48 ÷ 24 = 2
Constant ratio r = 2. This is geometric.
$

Example 2: 100, 50, 25, 12.5, ...

$50 ÷ 100 = 1/2, 25 ÷ 50 = 1/2, 12.5 ÷ 25 = 1/2
Constant ratio r = 1/2. This is geometric.
$

Example 3: 1, 4, 9, 16, 25, ... (square numbers)

$4 ÷ 1 = 4, 9 ÷ 4 = 2.25 — not constant. NOT geometric.
$

5. The nth Term of a Geometric Sequence

For a geometric sequence with first term a and common ratio r:

Tₙ = a × r^(n−1)

Where does this formula come from?

T₁ = a
T₂ = a × r
T₃ = a × r²
T₄ = a × r³
...
Tₙ = a × r^(n−1)     ← n−1 multiplications by r from the first term

Example 1: Find the 8th term of 2, 6, 18, 54, ...

$a = 2, r = 3
T₈ = 2 × 3^(8−1) = 2 × 3⁷ = 2 × 2,187 = 4,374
$

Example 2: Find the nth term of 64, 32, 16, 8, ...

$a = 64, r = 1/2
Tₙ = 64 × (1/2)^(n−1) = 64 × 2^(−(n−1)) = 2⁶ × 2^(1−n) = 2^(7−n)
$

So Tₙ = 2^(7−n). Check: n=1 → 2⁶ = 64 ✓, n=4 → 2³ = 8 ✓.

Example 3: The 3rd term of a geometric sequence is 36 and the 6th term is 972. Find a and r.

$T₃ = a × r² = 36
T₆ = a × r⁵ = 972

Divide: r⁵ / r² = 972 / 36
       r³ = 27
       r = 3

Then: a × 3² = 36 → a × 9 = 36 → a = 4

Tₙ = 4 × 3^(n−1)
$

6. The Fibonacci Sequence

The Fibonacci sequence is one of the most famous sequences in mathematics:

1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, ...

Rule: Each term (after the first two) is the sum of the two preceding terms.

F₁ = 1, F₂ = 1 Fₙ = Fₙ₋₁ + Fₙ₋₂ for n ≥ 3

Extending the sequence:

$F₁ = 1
F₂ = 1
F₃ = 1 + 1 = 2
F₄ = 1 + 2 = 3
F₅ = 2 + 3 = 5
F₆ = 3 + 5 = 8
F₇ = 5 + 8 = 13
F₈ = 8 + 13 = 21
... and so on
$

⚠️ THIS IS CRITICAL — The Fibonacci sequence appears throughout nature: spiral patterns in shells, branching in trees, arrangements of petals, and even galaxy structures. The ratio of consecutive Fibonacci terms approaches the golden ratio φ ≈ 1.618, which has profound connections to art, architecture, and aesthetics.

Practice: Find F₁₀ (the 10th Fibonacci number). 1, 1, 2, 3, 5, 8, 13, 21, 34, 55 F₁₀ = 55.

7. Other Famous Sequences

Triangular Numbers: 1, 3, 6, 10, 15, 21, 28, ...

$Tₙ = n(n + 1) / 2
$

Each term counts the dots in a triangular arrangement. T₁ = 1, T₂ = 3 (one on top, two below), T₃ = 6, etc.

Square Numbers: 1, 4, 9, 16, 25, 36, 49, ...

$Tₙ = n²
$

Cube Numbers: 1, 8, 27, 64, 125, ...

$Tₙ = n³
$

Powers of 2: 2, 4, 8, 16, 32, 64, ...

$Tₙ = 2ⁿ
$

(Note: some definitions start at 1 = 2⁰, but commonly adapted.)

8. Pattern Recognition — Finding the Rule

When given the first few terms of a sequence, how do you figure out the rule?

Step 1 — Check for arithmetic: Subtract consecutive terms. If the difference is constant, it's arithmetic.

Step 2 — Check for geometric: Divide consecutive terms. If the ratio is constant, it's geometric.

Step 3 — Check for "second difference" constant: If first differences aren't constant, check if the differences of the differences are constant. This indicates a quadratic pattern (Tₙ = an² + bn + c).

Example (quadratic): 2, 5, 10, 17, 26, ...

$First differences: 3, 5, 7, 9 (not constant)
Second differences:   2, 2, 2 (CONSTANT!)
This is a quadratic sequence: Tₙ = n² + 1
$

Step 4 — Look for known patterns: Triangular, Fibonacci, powers, etc.

Step 5 — Try to relate each term to its position n.

Example: Find the rule for 2, 6, 12, 20, 30, ...

$Position n:    1    2    3    4    5
Term:          2    6   12   20   30
Difference:      4    6    8   10 (first diff: 4, 6, 8, 10)
2nd diff:          2    2    2     (constant → quadratic)

Pattern insight: 2=1×2, 6=2×3, 12=3×4, 20=4×5, 30=5×6
Tₙ = n(n + 1)
$

9. Finding the Position Given the Value

Sometimes you know the term value and need to find n (its position in the sequence).

Example 1 (arithmetic): In the sequence 7, 11, 15, 19, ..., which term equals 87?

$Tₙ = 7 + (n − 1) × 4 = 7 + 4n − 4 = 4n + 3
Set Tₙ = 87:
4n + 3 = 87
4n = 84
n = 21
$

87 is the 21st term.

Example 2 (geometric): In the sequence 5, 15, 45, 135, ..., which term equals 3,645?

$Tₙ = 5 × 3^(n−1)
Set Tₙ = 3,645:
5 × 3^(n−1) = 3,645
3^(n−1) = 729
3^(n−1) = 3⁶
n − 1 = 6
n = 7
$

3,645 is the 7th term.

Worked Examples

Example 1: Arithmetic nth Term and Position

For the sequence 13, 18, 23, 28, ...: (a) Find the nth term formula. (b) Find the 50th term. (c) Which term equals 203?

Solution:

(a) a = 13, d = 5 Tₙ = 13 + (n − 1) × 5 = 13 + 5n − 5 = 5n + 8

(b) T₅₀ = 5(50) + 8 = 250 + 8 = 258

(c) Set Tₙ = 203: 5n + 8 = 203 5n = 195 n = 39 The 39th term.

Example 2: Geometric nth Term

For the sequence 3, 12, 48, 192, ...: (a) Find the nth term formula. (b) Find the 7th term.

Solution:

(a) a = 3, r = 12 ÷ 3 = 4 Tₙ = 3 × 4^(n−1)

(b) T₇ = 3 × 4⁶ = 3 × 4,096 = 12,288

Example 3: Pattern Recognition

Find the nth term of the sequence: 4, 9, 16, 25, 36, ...

Solution:

Check for arithmetic: 9−4=5, 16−9=7 → not constant. Check for geometric: 9÷4=2.25, 16÷9≈1.78 → not constant.

Look at the numbers: 2², 3², 4², 5², 6², ... These are square numbers starting from n=1 with T₁ = (1+1)² = 2².

Tₙ = (n + 1)² = n² + 2n + 1

Check: n=1 → 4 ✓, n=5 → 36 ✓.

Answer: Tₙ = (n + 1)²

Example 4: Visual Pattern to Sequence

A pattern of matchsticks forms squares end-to-end. One square uses 4 matchsticks. Each additional square uses 3 more matchsticks (sharing one side).

Square chain: □ □□ □□□ ... Matchsticks: 4, 7, 10, 13, ...

(a) Write the nth term for matchsticks. (b) How many matchsticks for 20 squares? (c) If you have 301 matchsticks, how many squares can you make?

Solution:

(a) a = 4, d = 3 Tₙ = 4 + (n − 1) × 3 = 4 + 3n − 3 = 3n + 1

(b) T₂₀ = 3(20) + 1 = 60 + 1 = 61 matchsticks

(c) Set Tₙ = 301: 3n + 1 = 301 3n = 300 n = 100 100 squares.

Key Terms

• For negative common differences or ratios.
• Fibonacci sequence
• arithmetic sequence
• Misidentifying Fibonacci-style sequences.
• Example 1: Arithmetic Nth Term And Position
• The 22nd term.
• Correct: B) Arithmetic with d = −5
• Correct: C) 5n + 2
• The Fibonacci Sequence
• Example 3: Pattern Recognition
• 12,288
• Answer: A) Arithmetic, d = 7

Practice Problems

(Answers are below. Try each problem before checking.)

Problem 1: Find the 15th term of the arithmetic sequence 8, 14, 20, 26, ...

Problem 2: Find the nth term of the arithmetic sequence 25, 19, 13, 7, 1, ...

Problem 3: In the arithmetic sequence 5, 11, 17, 23, ..., which term equals 131?

Problem 4: Find the 6th term of the geometric sequence 7, 21, 63, ...

Problem 5: Find the nth term of the geometric sequence 80, 40, 20, 10, ...

Problem 6: Find the nth term of the sequence 3, 8, 15, 24, 35, ...

Problem 7: Fibonacci: Find F₁₂ (the 12th Fibonacci number).

Summary

1. A sequence is an ordered list of numbers following a rule. An arithmetic sequence has a constant difference d between consecutive terms; a geometric sequence has a constant ratio r.
2. Arithmetic nth term: Tₙ = a + (n − 1)d, where a is the first term and d is the common difference.
3. Geometric nth term: Tₙ = a × r^(n−1), where a is the first term and r is the common ratio.
4. The Fibonacci sequence (1, 1, 2, 3, 5, 8, ...) adds the two preceding terms. It appears throughout nature and is connected to the golden ratio.
5. Pattern recognition strategy: Check arithmetic (first difference), then geometric (ratio), then second differences (quadratic), then known patterns (squares, triangular numbers, Fibonacci).
6. Finding the nth term is the foundation for algebraic functions — Tₙ expresses how the term depends on its position n.

Pitfalls

• Confusing arithmetic with geometric. Subtraction gives a constant difference → arithmetic. Division gives a constant ratio → geometric. Always test both.
• Using n instead of n−1 in formulas. Tₙ = a + nd is wrong — the first term should have n=1 giving a, not a+d.
• Forgetting the first difference in a quadratic sequence. If terms are 1, 4, 9, 16, the rule is Tₙ = n², not n² + something — always verify by substituting n=1.
• Misidentifying Fibonacci-style sequences. Not every sequence where terms are sums of previous terms is Fibonacci — check the specific recurrence relation.
• For negative common differences or ratios. d can be negative (decreasing arithmetic); r can be between 0 and 1 (decreasing geometric, like dividing by 2 each time).

Quiz

Answer each question, then read the explanation for your choice.

Q1: What type of sequence is 13, 8, 3, −2, −7, ...?

A) Geometric with r = −5 B) Arithmetic with d = −5 C) Arithmetic with d = 5 D) Neither

Q2: Find the nth term of 7, 12, 17, 22, ...

A) 7n + 5 B) 5n + 7 C) 5n + 2 D) 7n − 2

Q3: What is the 10th term of the geometric sequence 3, 6, 12, ...?

A) 60 B) 3,072 C) 1,536 D) 30

Q4: What is the next term in the Fibonacci sequence after 34?

A) 47 B) 53 C) 55 D) 42

Q5: Find the nth term of 2, 6, 12, 20, 30, ...

A) 2n B) n² + n C) n(n + 1) D) Both B and C

Q6: Which term of the sequence 11, 16, 21, 26, ... equals 106?

A) 19th B) 20th C) 21st D) 18th

Q7: For the geometric sequence 81, 27, 9, 3, ..., find Tₙ.

A) 81 × (1/3)ⁿ B) 81 / 3^(n−1) C) 81 × (1/3)^(n−1) D) Both B and C

Q8: What sequence is formed by triangular numbers?

A) 1, 2, 4, 8, 16, ... B) 1, 3, 6, 10, 15, ... C) 1, 2, 3, 5, 8, ... D) 1, 4, 9, 16, 25, ...

Next Steps

Move on to 00-14 — Factors, Multiples, and Primes (Deep Dive) for a comprehensive exploration of factor trees, HCF/LCM with three or more numbers, the Sieve of Eratosthenes, and prime factorization methods.

Q5: How many terms in the arithmetic sequence 2, 8, 14, ..., 92?

A) 14 B) 15 C) 16 D) 17

Answer: C) 16 Tₙ = 2 + (n−1)×6 = 6n − 4. Set 6n − 4 = 92 → 6n = 96 → n = 16.