MATHEMATICSAlgebra

CURRENT TOPICSequences & Patterns

NAVIGATION

3n+1

1, 4, 9, 16

● Unified Topic Guide Verified 11+ Curriculum

Sequences & The nᵗʰ Term Master Guide 🔍

A clean, structured 3-part guide covering number rules, double sequences, algebraic nᵗʰ term formulas, shape patterns, and exam strategies!

Practice Levels:Warm-Up Core Mode Pro Mode

Practice Papers

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PART 1: NUMBER SEQUENCES & PATTERNS

Understanding how sequence rules work and continuing number patterns.

Rule ➔ Pattern ➔ Next Term

1. HOW TO FIND THE RULE

A number sequence is an ordered list of numbers following a rule. Follow these 4 steps:

1Look at the numbers carefully.

2Find the pattern/difference between terms.

3Describe the rule in words (e.g., "add 3 each time").

4Use the rule to continue or find missing terms.

COMMON SEQUENCE RULES

+ Add	+ 3	(add 3 each time)
− Subtract	− 4	(subtract 4 each time)
× Multiply	× 2	(multiply by 2 each time)
÷ Divide	÷ 5	(divide by 5 each time)
Expanding Gap	+ 2, + 4, + 6, + 8...
Special Patterns	Square numbers (1,4,9,16), Fibonacci (1,1,2,3,5)

2. WORKED EXAMPLES – 8 COMMON PATTERNS

A2, 5, 8, 11, 14, __, __

Rule: add 3 each time

Next: 17, 20

B50, 45, 40, 35, 30, __, __

Rule: subtract 5 each time

Next: 25, 20

C3, 6, 12, 24, 48, __, __

Rule: multiply by 2 each time

Next: 96, 192

D81, 27, 9, 3, 1, __, __

Rule: divide by 3 each time

Next: 1/3, 1/9

E1, 3, 6, 10, 15, __, __

Rule: add 2, then 3, 4, 5, ...

Next: 21, 28

F2, 4, 8, 16, 32, __, __

Rule: multiply by 2 each time

Next: 64, 128

G5, 9, 17, 33, 65, __, __

Rule: multiply by 2, then -1

Next: 129, 257

H100, 95, 85, 70, 50, __, __

Rule: subtract 5, 10, 15, 20, ...

Next: 25, -5

3. DOUBLE SEQUENCES (EVEN VS ODD POSITIONED TERMS)

EVEN TERMS (2ⁿᵈ, 4ᵗʰ, 6ᵗʰ, 8ᵗʰ...)

Look at even-positioned terms to find the rule:

Sequence: 4, 7, 12, 13, 36, 19, 108...

7 ➔ 13 ➔ 19 (+6 each time)

Rule for even terms: add 6

What is the 8ᵗʰ term?25

ODD TERMS (1ˢᵗ, 3ʳᵈ, 5ᵗʰ, 7ᵗʰ...)

Look at odd-positioned terms to find the rule:

Sequence: 2, 4, 11, 8, 56, 16...

2 ➔ 11 ➔ 56 (×5 + 1 each time)

Rule for odd terms: multiply by 5, then add 1

What is the 7ᵗʰ term?281

4. PRACTICE: CONTINUE THE SEQUENCES

Write the next 3 terms (separated by commas, e.g. 31,36,41):

1) 6, 11, 16, 21, 26...

2) 100, 90, 81, 73, 66...

3) 1, 4, 9, 16, 25...

4) 2, 5, 10, 17, 26...

5) 3, 9, 27, 81, 243...

6) 50, 48, 44, 38, 30...

7) 2, 6, 4, 8, 6, 10...

8) 5, 12, 19, 26, 33...

PART 2: THE nᵀᴴ TERM RULE & FORMULAS

Using algebra to find any term in a sequence instantly without writing them all out.

Formula ➔ Position (n) ➔ Instant Answer

5. WHAT IS THE nᵀᴴ TERM?

Each number in a sequence is called a term. The nᵗʰ term represents the term at position n:

Position (n):	1	2	3	4	n
Term:	4	7	10	13	3n + 1

📌 n = position: 1ˢᵗ term is n=1, 2ⁿᵈ term is n=2, 100ᵗʰ term is n=100.

HOW TO FIND THE nᵀᴴ TERM FORMULA

1Find the difference between terms (e.g., +3 each time ➔ starts with 3n).

2Multiply 3 by position 1 (3 × 1 = 3).

3Adjust to match 1ˢᵗ term (3 to 2 ➔ subtract 1 ➔ 3n - 1).

4Test with n=2 to double check (3(2) - 1 = 5 ✓).

6. WORKED EXAMPLES – nᵀᴴ TERM FORMULAS

A. Counting on by 3

Sequence: 2, 5, 8, 11, 14, 17...

Difference: +3 ➔ start with 3n

nᵗʰ term rule: 3n - 1

Check n=1 ➔ 3(1) - 1 = 2 ✓

Check n=4 ➔ 3(4) - 1 = 11 ✓

Find 10ᵗʰ term: 3(10) - 1 ═ 29

B. Counting on by 4

Sequence: 5, 9, 13, 17, 21, 25...

Difference: +4 ➔ start with 4n

nᵗʰ term rule: 4n + 1

Check n=1 ➔ 4(1) + 1 = 5 ✓

Check n=5 ➔ 4(5) + 1 = 21 ✓

Find 15ᵗʰ term: 4(15) + 1 ═ 61

C. Counting back by 5

Sequence: 50, 45, 40, 35, 30...

Difference: -5 ➔ start with -5n

nᵗʰ term rule: 55 - 5n

Check n=1 ➔ 55 - 5(1) = 50 ✓

Check n=4 ➔ 55 - 5(4) = 35 ✓

Find 12ᵗʰ term: 55 - 5(12) ═ -5

7. COMPLETE THE FORMULA TABLES

a) Rule: 2n + 3

n = 10 ➔ Term =

b) Rule: 5n - 2

n = 5 ➔ Term =n = 20 ➔ Term =

c) Rule: 30 - 3n

n = 4 ➔n = 5 ➔n = 12 ➔

8. VISUAL SHAPE SEQUENCES

How many squares will be in the 6ᵗʰ shape?

Shape 11 sq

Shape 23 sq

Shape 35 sq

Pattern (in words): Add 2 squares each time.

nᵗʰ term rule: 2n - 1

6ᵗʰ shape: 2(6) - 1 ═ 11 squares

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PART 3: SUMMARY, TIPS & KNOWLEDGE CHECK

Exam tips, avoiding common traps, and testing your understanding.

Mastery & Exam Readiness

TOP TIPS

✓ Look for difference between numbers.
✓ Test with small values of n (n=1).
✓ Try even/odd terms for tricky sequences.
✓ Always test rule on whole sequence!

COMMON PATTERNS

• Add/subtract same number.
• Multiply/divide same number.
• Pattern in difference (+2, +4, +6).
• Pattern in multiplier (×2, ×3, ×2).

COMMON MISTAKES

❌ Assuming rule too quickly.
❌ Only checking part of sequence.
❌ Forgetting to test all numbers.
❌ Missing 2 rules (even/odd).

DETECTIVE CLUES 🔍

"Clues are everywhere! Look for patterns and use the rule to solve the mystery!"

9. CHECK YOUR UNDERSTANDING QUIZ

1) What is the 8ᵗʰ term of the sequence with nᵗʰ term rule 3n + 2?

2) What is the nᵗʰ term rule of the sequence 40, 36, 32, 28...?

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