Number Patterns and Sequences

Lesson 3 of 3: Patterns and Sequences

In this lesson:

• Identify the rule in any number sequence
• Form sequences from number types (prime, square, cubic, triangular)
• Predict any term directly using the nth-term formula

Learning Objectives for This Lesson

1. Identify the rule in a sequence — arithmetic, geometric, or type-based
2. Form and extend sequences from square, cubic, triangular, and prime numbers
3. Predict the nth term of an arithmetic sequence using
4. Find missing terms in a sequence
5. Determine which term has a given value

A Savings Group Problem to Solve

A savings group starts with 5,000 UGX. The pot grows by 3,000 UGX each week.

Week 1: 5,000 | Week 2: 8,000 | Week 3: 11,000 | Week 4: 14,000

Question: Without counting all 20 weeks — what is in the pot after week 20?

State the Rule in Words

Three sequences. Before any labels — what is the rule?

(a) 3, 6, 9, 12, 15, …

(b) 2, 4, 8, 16, 32, …

(c) 1, 4, 9, 16, 25, …

For each: describe the rule in words. Then predict the next term.

Write your descriptions before the next slide.

Three Types of Number Sequences

• Arithmetic: constant difference between consecutive terms
• Geometric: constant ratio between consecutive terms
• Type-based: terms are specific kinds of numbers (prime, triangular, square, cubic…)

How to Classify a Sequence Correctly

Rule: Check at least two consecutive differences (or ratios) before deciding.

For 2, 4, 8, 16:

• Differences: 4−2 = 2, 8−4 = 4 → not equal → NOT arithmetic
• Ratios: 4÷2 = 2, 8÷4 = 2 → equal → Geometric (ratio = 2)

Quick Check: Classify the Sequences

Classify each as arithmetic, geometric, or other. Give the rule.

1. 3, 7, 11, 15, 19, …
2. 2, 4, 8, 16, 32, …
3. 1, 1, 2, 3, 5, 8, 13, … (what type?)

For arithmetic sequences, state and .

Number-Type Sequences: Using Lesson 2

• Prime: 2, 3, 5, 7, 11, 13, …
• Triangular: 1, 3, 6, 10, 15, 21, …
• Cubic: 1, 8, 27, 64, 125, 216, …
• Composite: 4, 6, 8, 9, 10, 12, …
• Even: 2, 4, 6, 8, 10, …

Which of these are arithmetic?

Identify the Type and Extend

Identify the number type and give the next two terms:

(a) 2, 3, 5, 7, 11, 13, __, __

(b) 1, 3, 6, 10, 15, 21, __, __

(c) 8, 27, 64, 125, __, __

(d) 4, 6, 8, 9, 10, 12, __, __

Name the type and the rule.

Forming Sequences: Start from the Type

Given a number type, form the first 6 terms of its sequence:

1. Triangular numbers starting from
2. Odd numbers starting from 1
3. Square numbers starting from

Write out the first 6 terms for each.

Quick Check: Identify Sequence Types

For each sequence below, give the rule and the next two terms:

1. 5, 10, 15, 20, … (type?)
2. 1, 8, 27, 64, 125, … (type?)
3. 2, 3, 5, 7, 11, 13, … (type?)

For arithmetic sequences: state and .

Back to the Savings Group

How many times is 3,000 added in week n?

Deriving the nth-Term Formula Step by Step

• Term 1: (added zero times)
• Term 2: (added once)
• Term 3: (added twice)
• Term : (added exactly times)

Watch Out: The Off-by-One Trap

5th term of 2, 5, 8, 11, …?

(used , not )

Sequence: 2, 5, 8, 11, 14 — 5th term = 14.

✓

Check: set ; formula must give .

The nth-Term Formula Applied to Examples

For an arithmetic sequence with first term and common difference :

Savings group (, , ):

Sequence 5, 8, 11, … (, , ):

Warning: Formula Applies Only to Arithmetic Sequences

Sequence: 3, 6, 12, 24, … (geometric, ratio = 2)

Treat as arithmetic (, ): 10th term =

✓ Actual 10th term:

Differences not constant → NOT arithmetic → do not use the formula.

Finding Which Term Has a Given Value

Which term in 3, 6, 9, 12, … equals 93?

Set up:

Check: ✓ → 93 is the 31st term.

Is the Value in the Sequence?

Is 100 in 5, 8, 11, 14, …?

32.67 is not a whole number → 100 is NOT in this sequence.

Nearest: gives 98; gives 101.

Your Turn: Apply the nth-Term Formula

1. Which term in 7, 14, 21, 28, … equals 91?
2. Is 100 in the sequence 7, 14, 21, 28, …?

If n is not a whole number, the value is not in the sequence.

Practice: Find the 15th Term

For the sequence 7, 12, 17, 22, …:

1. State and
2. Find the 15th term using the formula
3. Verify by writing out the sequence to 5 terms and confirming the pattern

Show full working.

Key Takeaways: Sequence Types and Rules

✓ Arithmetic: constant difference — check two differences

✓ Geometric: constant ratio — not arithmetic

✓ Type-based: apply the number-type rule directly

Always classify before using any formula

Key Takeaways: Using the nth-Term Formula

✓ Formula: — use , not

✓ Find which term: set formula = value, solve for

✓ If is not a whole number, the value is not in the sequence

— your first algebraic generalisation from a pattern.