GCSE Maths Algebra - How to Solve: Sequences (nth Term, Quadratic, Special Sequences) – Complete Guide

How to Solve: Sequences (nth Term, Quadratic, Special Sequences) – Complete Guide

For GCSE & A-Level (Physics, Chemistry, Biology) – Ace Your Exam!

Introduction

"Mastering sequences unlocks 5–10% of your GCSE Maths paper—and the same skills help you predict reaction rates in Chemistry or genetic patterns in Biology. One wrong nth term formula could cost you 4–6 marks in a single question. Let’s fix that."

WHAT YOU NEED TO KNOW FIRST

1. Linear equations: You must be able to solve y = mx + c for x or y.
2. Substitution: Plugging numbers into formulas (e.g., n = 1, 2, 3).
3. Basic algebra: Expanding brackets (e.g., (n + 1)² = n² + 2n + 1).

KEY TERMS & FORMULAS

1. Linear Sequences (Arithmetic Sequences)

Formula (nth term): Tₙ = a + (n – 1)d - Tₙ = term at position n - a = first term - d = common difference (MEMORISE THIS) - n = term number (e.g., 1st, 2nd, 3rd…)

Example: Sequence: 3, 7, 11, 15… - a = 3, d = 4 - nth term: Tₙ = 3 + (n – 1) × 4 = 4n – 1

2. Quadratic Sequences

Formula (nth term): Tₙ = an² + bn + c - a, b, c = constants to find (MEMORISE METHOD, NOT FORMULA) - a = half the second difference (see Step 2 below)

Key Idea: - First differences: Differences between terms (e.g., 7–3 = 4). - Second differences: Differences of the first differences (must be constant for quadratic sequences).

3. Special Sequences

STEP-BY-STEP METHOD

For Linear Sequences (Arithmetic)

Step 1: Write the sequence and label terms T₁, T₂, T₃… Step 2: Find the common difference (d) by subtracting consecutive terms. Step 3: Write the nth term formula: Tₙ = a + (n – 1)d Step 4: Simplify the formula (e.g., 3 + (n – 1)×4 = 4n – 1).

Worked Example: Sequence: 5, 9, 13, 17… 1. T₁ = 5, T₂ = 9, T₃ = 13 2. d = 9 – 5 = 4 3. Tₙ = 5 + (n – 1)×4 4. Simplify: Tₙ = 5 + 4n – 4 = 4n + 1

For Quadratic Sequences

Step 1: Write the sequence and label terms T₁, T₂, T₃… Step 2: Calculate first differences (e.g., T₂ – T₁, T₃ – T₂…). Step 3: Calculate second differences (differences of first differences). If constant, it’s quadratic. Step 4: a = second difference ÷ 2. Step 5: Write Tₙ = an² + bn + c. Substitute n = 1, 2, 3 to form equations. Step 6: Solve for b and c using simultaneous equations.

Worked Example: Sequence: 2, 5, 10, 17, 26… 1. Terms: T₁ = 2, T₂ = 5, T₃ = 10, T₄ = 17, T₅ = 26 2. First differences: 3, 5, 7, 9 3. Second differences: 2, 2, 2 → a = 2 ÷ 2 = 1 4. Tₙ = n² + bn + c 5. Substitute n = 1: 1 + b + c = 2 → b + c = 1 (Equation 1)
Substitute n = 2: 4 + 2b + c = 5 → 2b + c = 1 (Equation 2) 6. Subtract Equation 1 from Equation 2: b = 0
Then c = 1
Final nth term: Tₙ = n² + 1

For Special Sequences

Step 1: Recognise the pattern (e.g., squares, cubes, triangular numbers). Step 2: Use the memorised formula (see table above). Step 3: Verify by substituting n = 1, 2, 3.

Worked Example: Sequence: 1, 3, 6, 10, 15… 1. Recognise as triangular numbers. 2. Formula: Tₙ = n(n + 1)/2 3. Check:
- n = 1: 1×2/2 = 1 ✔️
- n = 2: 2×3/2 = 3 ✔️

WORKED EXAMPLES

Example 1 – Basic Linear Sequence

Question: Find the nth term of 8, 11, 14, 17… Solution: 1. T₁ = 8, T₂ = 11, T₃ = 14 2. d = 11 – 8 = 3 3. Tₙ = 8 + (n – 1)×3 4. Simplify: Tₙ = 3n + 5

What we did and why: - Found d by subtracting terms. - Used the linear nth term formula. - Simplified to get the final answer.

Example 2 – Medium Quadratic Sequence

Question: Find the nth term of 3, 8, 15, 24, 35… Solution: 1. Terms: T₁ = 3, T₂ = 8, T₃ = 15, T₄ = 24, T₅ = 35 2. First differences: 5, 7, 9, 11 3. Second differences: 2, 2, 2 → a = 1 4. Tₙ = n² + bn + c 5. Substitute n = 1: 1 + b + c = 3 → b + c = 2 (Equation 1)
Substitute n = 2: 4 + 2b + c = 8 → 2b + c = 4 (Equation 2) 6. Subtract Equation 1 from Equation 2: b = 2
Then c = 0
Final nth term: Tₙ = n² + 2n

What we did and why: - Checked second differences to confirm it’s quadratic. - Used a = second difference ÷ 2. - Solved for b and c using simultaneous equations.

Example 3 – Exam-Style (Disguised)

Question: A sequence starts 4, 13, 26, 43… Find the 10th term. Solution: 1. Terms: T₁ = 4, T₂ = 13, T₃ = 26, T₄ = 43 2. First differences: 9, 13, 17 3. Second differences: 4, 4 → a = 2 4. Tₙ = 2n² + bn + c 5. Substitute n = 1: 2 + b + c = 4 → b + c = 2 (Equation 1)
Substitute n = 2: 8 + 2b + c = 13 → 2b + c = 5 (Equation 2) 6. Subtract Equation 1 from Equation 2: b = 3
Then c = -1
Final nth term: Tₙ = 2n² + 3n – 1 7. 10th term: T₁₀ = 2(10)² + 3(10) – 1 = 200 + 30 – 1 = 229

What we did and why: - Recognised it’s quadratic from second differences. - Solved for constants b and c. - Substituted n = 10 to find the specific term.

COMMON MISTAKES

1. MISTAKE: Forgetting to subtract 1 in linear nth term (Tₙ = a + nd instead of Tₙ = a + (n – 1)d).
   WHY IT HAPPENS: Misremembering the formula.
   CORRECT APPROACH: Always write Tₙ = a + (n – 1)d first, then simplify.

2. MISTAKE: Assuming all sequences are linear.
   WHY IT HAPPENS: Not checking second differences.
   CORRECT APPROACH: Calculate first differences first. If they’re not constant, check second differences.

3. MISTAKE: Incorrectly calculating a for quadratic sequences (a = second difference instead of a = second difference ÷ 2).
   WHY IT HAPPENS: Confusing the rule.
   CORRECT APPROACH: a is always half the second difference.

4. MISTAKE: Mixing up n and Tₙ in substitution.
   WHY IT HAPPENS: Careless labelling.
   CORRECT APPROACH: Write T₁ = … before substituting.

5. MISTAKE: Not simplifying the final nth term.
   WHY IT HAPPENS: Rushing.
   CORRECT APPROACH: Always expand and simplify (e.g., 3 + 4(n – 1) = 4n – 1).

EXAM TRAPS

1. TRAP: The sequence is not starting at n = 1.
   HOW TO SPOT IT: Question says "the first term is when n = 0" or "the sequence starts at n = 2".
   HOW TO AVOID IT: Adjust the formula. If n starts at 0, use Tₙ = a + nd. If n starts at 2, substitute n = 2 first.

2. TRAP: The sequence is recursive (e.g., Fibonacci).
   HOW TO SPOT IT: Terms depend on previous terms (e.g., "each term is the sum of the two before it").
   HOW TO AVOID IT: Don’t try to find an nth term—use the recursive rule directly.

3. TRAP: The sequence is disguised (e.g., "the number of diagonals in an n-sided polygon").
   HOW TO SPOT IT: The question describes a real-world pattern, not just numbers.
   HOW TO AVOID IT: Write out the first few terms, then treat it like a normal sequence.

1-MINUTE RECAP

"Here’s what you need to remember tonight: 1. Linear sequences: Find d, use Tₙ = a + (n – 1)d, simplify. 2. Quadratic sequences: Check second differences, a = second difference ÷ 2, solve for b and c. 3. Special sequences: Memorise squares (n²), cubes (n³), and triangular numbers (n(n + 1)/2). 4. Avoid traps: Check if n starts at 1, watch for recursive rules, and always simplify your answer. 5. Practice: Do one linear and one quadratic question before bed. You’ve got this!"