Sequences, Surds and Sets
๐ SEQUENCES, SURDS & SETS
Cambridge IGCSE Mathematics
๐ 9.1 SEQUENCES
Definition & Notation
Sequence: A list of numbers following a specific rule. Each number = term.
- T₁, T₂, T₃ = 1st, 2nd, 3rd terms
- Tโ = nth term (general term)
Types of Sequences
| Type | Rule | Example | Formula |
|---|---|---|---|
| Arithmetic | Common diff (d) | 2,6,10,14 | Tโ=dn+c |
| Geometric | Common ratio (r) | 3,6,12,24 | Tโ=ar^(n-1) |
| Quadratic | 2nd diff constant | 2,5,10,17 | Tโ=an²+bn+c |
| Square | n² | 1,4,9,16 | Tโ=n² |
| Triangular | Sum naturals | 1,3,6,10 | Tโ=½n(n+1) |
Finding nth Term (Arithmetic)
METHOD:
1. Find common difference (d)
2. Start with: dn
3. Adjust by comparing
Example: 5, 8, 11, 14, ...
d = 3, start with 3n
When n=1: 3(1)=3, term=5
Difference: +2
Answer: Tโ = 3n + 2
Finding nth Term (Quadratic)
METHOD:
1. Check 2nd diff constant
2. If 2nd diff = 2a → formula has an²
3. Compare n² with terms
Example: 2, 5, 10, 17, 26
1st diff: 3,5,7,9 (not constant)
2nd diff: 2,2,2 (constant!)
Compare n²: 1,4,9,16,25
Each term = n² + 1
Answer: Tโ = n² + 1
⚡ Quick Check:
To verify if number is in sequence:
- Set Tโ = that number
- Solve for n
- If n is whole → YES
- If n not whole → NO
๐ข 9.2 RATIONAL & IRRATIONAL
Definitions
| Type | Definition | Examples |
|---|---|---|
| Rational | Can write as a/b | 5, 0.5, 0.333... |
| Irrational | Cannot write as fraction | ฯ, √2, √3, √5 |
Note: √4=2, √9=3, √16=4 are RATIONAL!
Types of Decimals
- Terminating: 0.5, 0.125 → Rational ✓
- Recurring: 0.333..., 0.45̇45̇ → Rational ✓
- Non-recurring: √2=1.414... → Irrational ✗
Recurring Notation:
- 0.333... = 0.3̇
- 0.454545... = 0.4̇5̇
- 0.583333... = 0.583̇
Converting Recurring to Fractions
METHOD:
1. Let x = recurring decimal
2. Multiply by 10โฟ (n=repeating digits)
3. Subtract equations
4. Solve for x
Example 1: 0.6̇ = ?
Let x = 0.666...
10x = 6.666...
10x - x = 6
9x = 6
x = 6/9 = 2/3
Example 2: 0.2̇4̇ = ?
Let x = 0.242424...
100x = 24.242424...
100x - x = 24
99x = 24
x = 24/99 = 8/33
๐ก Rule: n digits repeat → ×10โฟ
√ 9.3 SURDS
Definition
Surd: Exact root value that can't be simplified to rational number.
| SURDS | NOT SURDS |
|---|---|
| √2, √3, √5, √7, √8 | √4=2, √9=3, √16=4 |
Surd Rules
RULES:
√(x×y) = √x × √y
√(x÷y) = √x ÷ √y
(√x)² = x
√x × √x = x
Perfect Squares:
1²=1 2²=4 3²=9 4²=16
5²=25 6²=36 7²=49
8²=64 9²=81 10²=100
Simplifying Surds
Find perfect square factors
Example 1: √50
√50 = √(25×2) = 5√2
Example 2: √18
√18 = √(9×2) = 3√2
Example 3: 2√18
2√18 = 2×3√2 = 6√2
Operations with Surds
Adding/Subtracting
3√5 + 5√5 = 8√5
7√5 - √20 = 7√5 - 2√5 = 5√5
Multiplying
√3 × √7 = √21
2√3 × 3√5 = 6√15
√3 × √12 = √36 = 6
Dividing
√21 ÷ √3 = √7
8√30 ÷ 2√6 = 4√5
Expanding
√5(√2+3) = √10 + 3√5
(√5+2)(√5-3) = 5-3√5+2√5-6
= -1-√5
Rationalising Denominator
Remove surd from bottom
One term:
5/√3 = (5√3)/(√3×√3)
= 5√3/3
Two terms (use conjugate):
3/(2+√5) = 3(2-√5)/[(2+√5)(2-√5)]
= 3(2-√5)/(4-5)
= 3(2-√5)/(-1)
= 3√5 - 6
Conjugate: (a+√b)(a-√b) = a²-b
๐️ 9.4 SETS
Set Notation
| Symbol | Meaning | Example |
|---|---|---|
| { } | Define set | A={1,2,3} |
| ∈ | element of | 3 ∈ A |
| ∉ | NOT element | 5 ∉ A |
| n(A) | number elements | n(A)=3 |
| ∅ | empty set | { } |
| โฐ | universal set | all elements |
| A' | complement | not in A |
| ⊆ | subset | B ⊆ A |
Set Operations
| Operation | Symbol | Meaning |
|---|---|---|
| Union | A ∪ B | All from both |
| Intersection | A ∩ B | Common only |
| Complement | A' | Not in A |
KEY FORMULAS:
n(A∪B) = n(A) + n(B) - n(A∩B)
n(โฐ) = n(A) + n(A')
A ∪ A' = โฐ
A ∩ A' = ∅
Set Builder Notation
Format: {x : condition}
{x : x∈primes, 10<x<20}
= {11, 13, 17, 19}
{x : x is integer, 0<x<20}
= {1, 2, 3, ..., 19}
Venn Diagrams
Rules:
- Rectangle = Universal (โฐ)
- Circles = Sets
- Overlap = Intersection
- All circles = Union
A = {1,2,3,4}
B = {3,4,5,6}
A∩B = {3,4} (overlap)
A∪B = {1,2,3,4,5,6}
(A∪B)' = {7}
๐ FORMULA REFERENCE
SEQUENCES:
Arithmetic: Tโ = a+(n-1)d or dn+c
Geometric: Tโ = ar^(n-1)
Quadratic: Tโ = an²+bn+c
Triangular: Tโ = ½n(n+1)
Square: Tโ = n²
SURDS:
√(x×y) = √x × √y
√(x÷y) = √x ÷ √y
(√x)² = x
Conjugate: (a+√b)(a-√b) = a²-b
SETS:
n(A∪B) = n(A)+n(B)-n(A∩B)
A ∪ A' = โฐ
A ∩ A' = ∅
⚠️ COMMON MISTAKES
❌ Sequences: Check n is whole number
❌ Decimals: Align repeating digits
❌ Surds: √(a+b) ≠ √a + √b
❌ Sets: No repeats, order doesn't matter
❌ Venn: Draw rectangle for โฐ
๐ฏ EXAM TIPS
✓ Show all working
✓ Use exact values (surds)
✓ Check by substituting back
✓ Simplify surds fully
✓ Label Venn diagrams with โฐ
✓ Write 0.3̇ not 0.3...
๐ END OF NOTES
Cambridge IGCSE Math
Chapter 9: Sequences, Surds & Sets
๐ Practice to master! ๐
