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Sequences & nth Term Tricks for O Level & IGCSE Mathematics

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Understanding Sequences as a High-Frequency Exam Topic

  • Sequences appear repeatedly in Paper 1 and Paper 2/4

  • Examiners love testing:

    • Recognising patterns

    • Finding next terms

    • Deriving the nth term

    • Using the nth term to find unknowns

    • Arithmetic vs non-arithmetic logic

  • Students lose marks mainly due to:

    • Skipping difference checks

    • Incorrect substitution

    • Confusing geometric and linear rules

    • Mixing up term number with term value

    • Forgetting zero position (term 0 vs term 1)

  • Mastering sequences strengthens skills in algebra, functions and modelling

Arithmetic Sequences: Core Tricks Every Student Must Memorise

  • Arithmetic sequence = sequence with constant difference

  • Common difference (d):

    • d = term₂ − term₁

    • d = term₃ − term₂

  • nth term formula:

    • a + (n − 1)d

    • a = first term

    • d = difference

  • Recognise arithmetic pattern quickly by checking differences

  • Example: 4, 7, 10, 13

    • d = 3

    • nth term = 4 + (n − 1)3 = 3n + 1

Fast Detection Hacks

  • If differences not equal → sequence is NOT arithmetic

  • If differences equal → nth term is linear

  • nth term graph is straight line

  • d tells slope

  • a tells vertical shift

Common Errors to Avoid

  • Using wrong difference sign

  • Forgetting to multiply d with (n − 1)

  • Writing formula as a + nd (missing −1)

  • Substituting n incorrectly

Finding the nth Term of Any Arithmetic Sequence Quickly

  • Step 1: Find difference d

  • Step 2: Write general form: Tₙ = dn + c

  • Step 3: Substitute n = 1 and T₁ to solve c

  • Step 4: Write final formula cleanly

  • Step 5: Test formula for term 2 or term 3

Shortcut Method (Fastest)

  • nth term = (difference × n) + (first term − difference)

  • Example: 11, 15, 19

    • d = 4

    • nth term = 4n + (11 − 4) = 4n + 7

Using nth Term to Solve Exam-Style Problems

Finding a Specific Term

  • Use formula directly

  • Example: T₂₀ = a + 19d

  • Substitute values carefully

Checking Whether a Number is in the Sequence

  • Set nth term equal to target value

  • Solve for n

  • If n is positive integer → number belongs

  • If n non-integer → number NOT in sequence

Finding Term Position

  • Solve Tₙ = given value

  • Rearrange equation for n

  • Provide integer-only answer

Geometric Sequences: Recognising When Multiplication Is Used

  • Geometric sequence uses constant ratio, not difference

  • Ratio r = term₂ ÷ term₁

  • nth term formula:

    • Tₙ = a × r⁽ⁿ⁻¹⁾

  • If each term multiplied by same number → geometric

  • Common exam examples:

    • Doubling patterns

    • Halving patterns

    • Growth models

    • Decay models

    • Repeated scaling

Fast Checks for Geometric Pattern

  • Differences NOT equal

  • Ratios equal

  • Sequence grows or shrinks rapidly

Students’ Most Common Mistakes

  • Using addition instead of multiplication

  • Using difference instead of ratio

  • Forgetting exponent (n − 1)

  • Applying negative ratio incorrectly

Quadratic Sequences: Detecting Curved Patterns

  • Quadratic sequences have second difference constant

  • First difference changes

  • Second difference = 2a in formula an² + bn + c

Step-by-Step Recognition

  • Step 1: Compute 1st differences

  • Step 2: Compute 2nd differences

  • Step 3: If 2nd difference constant → sequence quadratic

Finding nth Term of Quadratic

  • Write Tₙ = an² + bn + c

  • Use method:

    • 2a = second difference → find a

    • Substitute values of n = 1, 2, 3 to form equations

    • Solve for b and c

Quick Quadratic Shortcut

  • nth term for sequence starting like:

    • 2, 5, 10, 17, …

  • 1st difference: 3, 5, 7 → pattern odd numbers

  • nth term often matches n² + term behaviour

Special Sequences Examiners Love to Test

Fibonacci-Type Sequences

  • Each term = sum of previous two

  • Example: 1, 1, 2, 3, 5, 8

  • No formula needed

  • Just compute terms

Alternating Sequences

  • Pattern switches each step

  • Example: 2, −4, 2, −4,…

Two-Part Rules

  • Separate rule for odd and even positions

  • Example:

    • odd terms: increase by 3

    • even terms: multiply by 2

Piecewise Sequences

  • nth term changes based on condition

    • If n even → formula A

    • If n odd → formula B

Recognising Fake Patterns (Trick Questions)

  • Some sequences look arithmetic but aren’t

  • Some appear random but have piecewise rules

  • Always check differences FIRST

  • Avoid assuming arithmetic unless proven

  • Watch out for:

    • Alternating signs

    • Alternating differences

    • Mixed operations

    • Misleading numbers

Sequence Tables & Diagrams Used in Exams

  • Values sometimes shown in table form

  • You must detect:

    • Linear pattern

    • Quadratic pattern

    • Geometric pattern

    • Recurrence relation

  • Always compute differences before deciding

Using Graphs to Understand Sequences

  • Arithmetic → straight-line graph

  • Geometric → exponential shape

  • Quadratic → parabola shape

  • Mixed sequences → irregular plot

  • Use graph to confirm formula correctness

nth Term in Real-Life Exam Problems

Pattern Growth Questions

  • Squares in pattern

  • Matches in pattern

  • Tiles in geometric design

  • Perimeter or area growing with stage number

Strategy

  • List first few terms

  • Identify type of sequence

  • Find nth term

  • Apply to large n value

Solving Reverse Problems: When Given nth Term, Find Sequence

  • Simply substitute n = 1, 2, 3…

  • Generate first few terms

  • Identify pattern

  • Example:

    • Tₙ = 5n − 2

    • Terms: 3, 8, 13, 18… (d = 5)

Simultaneous Equations in nth Term Problems

Used especially in quadratic sequences.

  • T₁ = a + b + c

  • T₂ = 4a + 2b + c

  • T₃ = 9a + 3b + c

  • Solve system to find a, b, c

Harder Exam Questions: Combining Patterns

Hybrid Sequences

  • Example: first difference itself geometric

  • Example: alternating quadratic + linear

  • Example: formula changes after certain n

Approach

  • Look at first 5–7 terms

  • Analyse differences deeply

  • Identify mixed behaviour

  • Write separate nth terms if needed

Working with Recurrence Relations (Higher Paper)

  • Formula given as

    • Tₙ₊₁ = something involving Tₙ

  • Compute terms one by one

  • Example:

    • T₁ = 4

    • Tₙ₊₁ = 3Tₙ − 2

  • Keep track carefully

Avoiding the Biggest Student Mistakes

  • Assuming arithmetic when it’s not

  • Using the wrong n (e.g., plugging n = term value)

  • Forgetting (n − 1) in formulas

  • Computing difference incorrectly

  • Treating geometric ratio as difference

  • Losing marks for wrong sign

  • Not checking formula using T₂ or T₃

  • Reading question incorrectly

  • Writing term number instead of formula

  • Misunderstanding alternating sequences

  • Incorrect substitution for large n

Exam Techniques for Securing A*

  • Always check difference BEFORE writing formula

  • If second difference constant → quadratic

  • If ratio constant → geometric

  • Test nth term using first 3 terms

  • Use clean algebra for b and c in quadratic tasks

  • For large n questions, substitute carefully

  • For pattern problems, write out 5–6 terms

  • When stuck, try:

    • Table of values

    • Difference table

    • Ratio test

  • Check every answer with substitution

  • Use clear labeling T₁, T₂, T₃…

  • Keep all working neat and methodical

A* Habits for Sequence Excellence

  • Train yourself to spot patterns instantly

  • Build difference tables quickly

  • Memorise all nth term formulas

  • Double-check everything with substitution

  • Avoid skipping steps

  • Write clean structure for quadratic nth term

  • Develop instinct for alternating patterns

  • Practise both arithmetic and geometric reasoning

  • Always simplify nth term expression

  • Use strategic checking:

    • n = 1

    • n = 2

    • n = 3

Final Mastery Checklist

  • Check difference

  • Identify type

  • Write nth term

  • Verify using substitution

  • Solve required term

  • Use nth term to answer story problems

  • Present clean, simplified answer

  • Cross-check with term pattern

  • Ensure expression matches sequence behaviour

Hunain Zia

Hunain Zia has previously achieved considerable success in the high-school educational stream, gaining a massive 154 Total A Grades and 7 Major Distinctions, a process in which he broke/ set 11 different world records, including the most A grades ever achieved, most subjects ever appeared in, most distinctions, gaining distinctions across two separate boards (Pearson Edexcel and CAIE) within the same year/ ever, and even 19 hours of constant examination. All records stand intact today. Hunain pursued his Honors Accounting Degree from the prestigious Bentley University, and then completed an LLB (in Honors) from University of London. Currently, he manages multiple social and commercial projects, has founded digital streams and works tirelessly in the education sector.

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