Sequences

Overview

Sequences are a fundamental concept in mathematics, testing your ability to identify patterns, formulate algebraic rules, and predict future terms. For the OCR GCSE exam, this topic is a reliable source of marks if you master the core techniques. It involves understanding the structure of different types of progressions, from simple linear sequences to more complex quadratic and geometric series. A strong grasp of sequences not only prepares you for direct questions but also builds foundational skills for algebra, graph work, and problem-solving across the entire specification. Examiners will test your ability to move from a term-to-term rule to a position-to-term rule (the nth term) and apply these rules in various contexts.

Key Concepts

Concept 1: Linear Sequences

A linear sequence is a series of numbers where the difference between any two consecutive terms is constant. This is known as the common difference. The key skill is finding the nth term, which is always in the form an + b.

• Finding 'a': The coefficient 'a' is simply the common difference. If the sequence goes up by 4 each time, the rule starts with 4n.
• Finding 'b': To find the adjustment 'b', compare your 'an' sequence (e.g., the 4 times table) with the actual sequence. The difference between them is your value for 'b'. A useful shortcut is to find the '0th term' by stepping back from the first term.

Example: For the sequence 7, 10, 13, 16...

• The common difference is +3, so the rule starts with 3n.
• The 3 times table is 3, 6, 9... Our sequence is 7, 10, 13... To get from 3 to 7, we add 4. To get from 6 to 10, we add 4.
• The adjustment is +4. So, the nth term is 3n + 4.

Concept 2: Quadratic Sequences (Higher Tier Only)

A quadratic sequence has an n² term in its rule. The first differences between terms are not constant, but the second differences are. The general form of the nth term is an² + bn + c.

• Finding 'a': The coefficient 'a' is always half the value of the constant second difference. If the second difference is 4, 'a' is 2.
• Finding 'bn + c': Once you have your an² term, subtract it from the original sequence. The remaining numbers will form a new linear sequence. You then find the nth term of this new linear sequence to get your 'bn + c' part.

Example: For the sequence 3, 8, 17, 30, 47...

• First differences: +5, +9, +13, +17
• Second differences: +4, +4, +4
• The second difference is 4, so 'a' = 4 / 2 = 2. The rule starts with 2n².
• Subtract 2n² from the original sequence:
  • Term 1: 3 - 2(1)² = 1
  • Term 2: 8 - 2(2)² = 0
  • Term 3: 17 - 2(3)² = -1
• The remaining linear sequence is 1, 0, -1... which has the rule -n + 2.
• Combine the parts: The final nth term is 2n² - n + 2.

Concept 3: Special Sequences

• Fibonacci-type sequences: A sequence where each term is the sum of the two preceding terms. The exam question will always define this rule for you. Example: If the first two terms are 2 and 5, the sequence is 2, 5, 7, 12, 19...
• Geometric Progressions: A sequence where you multiply each term by a constant value, the common ratio, to get the next term. The nth term is given by arⁿ⁻¹, where 'a' is the first term and 'r' is the common ratio. Example: 2, 6, 18, 54... (common ratio is 3). The nth term is 2 x 3ⁿ⁻¹.

Mathematical/Scientific Relationships

• Linear nth Term Formula: T(n) = an + b

  • T(n): The value of the nth term
  • a: The common difference
  • n: The term's position number
  • b: The '0th term' or adjustment value
  • Must memorise

• Quadratic nth Term Formula: T(n) = an² + bn + c

  • a: Half the second difference
  • bn + c: The linear sequence remaining after subtracting an²
  • Must memorise (Higher Tier)

• Geometric nth Term Formula: T(n) = arⁿ⁻¹

  • a: The first term
  • r: The common ratio
  • n: The term's position number
  • Given on formula sheet (for some specifications, but good to know)

Practical Applications

Sequences appear in many real-world scenarios, making them a powerful mathematical tool. They are used in:

• Finance: Calculating compound interest or loan repayments involves geometric progressions.
• Computer Science: Algorithms for searching and sorting often rely on sequence-based logic.
• Biology: Modelling population growth can use sequence formulas.
• Art & Nature: The Fibonacci sequence is famously observed in the patterns of snail shells, flower petals, and pinecones.