Series (Copy)
SERIES — O LEVEL & IGCSE ADDITIONAL MATHEMATICS FORMULA SHEET + COMMON MISTAKES
| Topic | Formula / Rule | Common Mistake |
|---|---|---|
| Sequence | Ordered list of terms | Mixing with series |
| Series | Sum of sequence terms | Writing terms only |
| Arithmetic Sequence | Constant difference | Using ratio instead |
| Common Difference | d = Tₙ − Tₙ₋₁ | Subtracting wrong way |
| nth Term of AP | Tₙ = a + (n − 1)d | Using n instead of n − 1 |
| Sum of AP | Sₙ = n/2[2a + (n − 1)d] | Forgetting brackets |
| Alternative AP Sum Formula | Sₙ = n/2(a + l) | Mixing l and d |
| Last Term of AP | l = a + (n − 1)d | Using n instead of n − 1 |
| Geometric Sequence | Constant ratio | Using difference |
| Common Ratio | r = Tₙ/Tₙ₋₁ | Subtracting terms |
| nth Term of GP | Tₙ = arⁿ⁻¹ | Wrong power |
| Sum of GP | Sₙ = a(1 − rⁿ)/(1 − r) | Sign errors |
| Alternative GP Formula | Sₙ = a(rⁿ − 1)/(r − 1) | Mixing versions |
| Infinite GP Sum | S∞ = a/(1 − r) | Using when |r| ≥ 1 |
| Infinite GP Condition | |r| < 1 | Ignoring restriction |
| Arithmetic Mean | Average of two terms | Using geometric mean |
| Arithmetic Mean Formula | AM = (a + b)/2 | Dividing wrongly |
| Geometric Mean | √ab | Adding instead |
| Sigma Notation | Σ means summation | Ignoring limits |
| Sum of Natural Numbers | Σn = n(n + 1)/2 | Forgetting brackets |
| Sum of Squares | Σn² = n(n + 1)(2n + 1)/6 | Missing denominator |
| Sum of Cubes | Σn³ = [n(n + 1)/2]² | Forgetting square |
| Recursive Formula | Tₙ₊₁ based on previous term | Using nth formula incorrectly |
| Convergence | Sequence approaches value | Assuming always converges |
| Divergence | Sequence does not settle | Thinking oscillation converges |
| Increasing Sequence | Terms grow larger | Mixing with positive sequence |
| Decreasing Sequence | Terms reduce | Confusing with negative |
| Arithmetic Progression | Linear growth | Using exponential formula |
| Geometric Progression | Exponential growth | Using AP formula |
| Sum to Infinity | GP only | Using for AP |
| Decimal Recurrence | Convert recurring decimals carefully | Wrong algebra setup |
| Sigma Expansion | Expand carefully | Missing terms |
| Proof by Induction Base Case | Verify n = 1 first | Skipping verification |
| Induction Step | Assume true for k | Assuming answer directly |
| Telescoping Series | Terms cancel systematically | Expanding unnecessarily |
| Alternating Series | Signs alternate | Ignoring negative terms |
Written and Compiled By Sir Hunain Zia (AYLOTI), World Record Holder With 154 Total A Grades, 11 World Records and 7 Distinctions, Educate A Change.
QUICK ARITHMETIC PROGRESSION (AP) FORMULAS
| Formula | Expression |
|---|---|
| nth Term | Tₙ = a + (n − 1)d |
| Sum Formula | Sₙ = n/2[2a + (n − 1)d] |
| Alternative Sum Formula | Sₙ = n/2(a + l) |
| Last Term | l = a + (n − 1)d |
QUICK GEOMETRIC PROGRESSION (GP) FORMULAS
| Formula | Expression |
|---|---|
| nth Term | Tₙ = arⁿ⁻¹ |
| Sum Formula | Sₙ = a(1 − rⁿ)/(1 − r) |
| Infinite Sum | S∞ = a/(1 − r) |
| Convergence Condition | |r| < 1 |
Written and Compiled By Sir Hunain Zia (AYLOTI), World Record Holder With 154 Total A Grades, 11 World Records and 7 Distinctions, Educate A Change.
QUICK SIGMA FORMULAS
| Formula | Expression |
|---|---|
| Sum of Integers | Σn = n(n + 1)/2 |
| Sum of Squares | Σn² = n(n + 1)(2n + 1)/6 |
| Sum of Cubes | Σn³ = [n(n + 1)/2]² |
QUICK AP vs GP TABLE
| Feature | AP | GP |
|---|---|---|
| Pattern | Add/subtract | Multiply/divide |
| Constant | Difference d | Ratio r |
| Growth Type | Linear | Exponential |
| Example | 2, 5, 8, 11 | 2, 6, 18, 54 |
COMMON SERIES TYPES
| Type | Example |
|---|---|
| Arithmetic | 3, 7, 11, 15 |
| Geometric | 2, 4, 8, 16 |
| Harmonic | 1, 1/2, 1/3 |
| Alternating | 1, −1, 1, −1 |
| Fibonacci | 1, 1, 2, 3, 5 |
Written and Compiled By Sir Hunain Zia (AYLOTI), World Record Holder With 154 Total A Grades, 11 World Records and 7 Distinctions, Educate A Change.
EXAM HACKS
| Situation | Fast Trick |
|---|---|
| AP question | Look for constant difference |
| GP question | Look for constant ratio |
| Infinite sum | Check if |r| < 1 |
| nth term | Start counting from n − 1 |
| Sigma notation | Expand first few terms |
| AP sum | Use formula with least unknowns |
| GP equation | Divide consecutive terms |
| Recurring decimal | Let x = decimal first |
| Large powers in GP | Use calculator carefully |
| Convergence | Check ratio size |
MOST COMMON LOSSES OF MARKS
| Mistake | Why Marks Are Lost |
|---|---|
| Using AP formula for GP | Entire answer incorrect |
| Wrong common ratio | Wrong sequence type |
| Forgetting n − 1 | nth term incorrect |
| Infinite sum with |r| ≥ 1 | Invalid answer |
| Sign mistakes in GP formula | Wrong totals |
| Missing brackets | Algebra errors |
| Wrong sigma expansion | Incorrect sums |
| Mixing sequence and series | Formula misuse |
| Arithmetic errors in induction | Proof collapses |
| Incorrect recursive substitution | Wrong term generation |
Written and Compiled By Sir Hunain Zia (AYLOTI), World Record Holder With 154 Total A Grades, 11 World Records and 7 Distinctions, Educate A Change.