EDEXCEL GCSE HIGHER MATHS - What are Sequences (Nth term and Quadratic Sequences)

EDEXCEL GCSE Higher Maths: Sequences (Nth Term and Quadratic Sequences)

Introduction

Sequences are a fundamental concept in mathematics. They are a list of numbers arranged in a specific order, where each number in the sequence is determined by a rule or pattern. Understanding sequences, particularly the nth term and quadratic sequences, is essential for GCSE Higher Maths.

Nth Term

The nth term of a sequence is a formula that allows you to find any term in the sequence without having to work out all the previous terms. This formula is usually expressed in terms of 'n', where 'n' represents the position of the term in the sequence.

Finding the Nth Term of a Linear Sequence

• Step 1: Identify the common difference between consecutive terms. This is the amount by which the sequence increases or decreases.
• Step 2: Determine the value of the first term.
• Step 3: Use the formula: nth term = (common difference * n) + (first term - common difference)

Example:

Consider the sequence: 2, 5, 8, 11...

• Common difference = 3
• First term = 2
• Nth term = (3 * n) + (2 - 3) = 3n - 1

Finding the Nth Term of a Non-Linear Sequence

For sequences that are not linear, finding the nth term requires more advanced techniques, such as:

• Recognising patterns: Look for relationships between the terms, such as squares, cubes, or factorials.
• Using differences: Calculate the differences between consecutive terms. If the differences are not constant, calculate the differences of the differences, and so on. This process can help you identify the pattern and the nth term.

Example:

Consider the sequence: 1, 4, 9, 16...

• This sequence is the sequence of squares: 1², 2², 3², 4²
• Therefore, the nth term is n²

Quadratic Sequences

A quadratic sequence is a sequence where the second differences are constant. This means that the difference between consecutive terms increases by a fixed amount.

Finding the Nth Term of a Quadratic Sequence

• Step 1: Calculate the first and second differences.
• Step 2: The coefficient of the n² term is half the second difference.
• Step 3: Substitute the value of the first term, the second term, and the coefficient of the n² term into the formula: nth term = (coefficient of n²) * n² + (coefficient of n) * n + constant.
• Step 4: Solve for the coefficients of n and the constant term.

Example:

Consider the sequence: 2, 7, 14, 23...

• First differences: 5, 7, 9
• Second differences: 2, 2
• Coefficient of n² = 2/2 = 1
• Using the formula: nth term = n² + (coefficient of n) * n + constant
• Substituting the values of the first and second terms:
  • 2 = 1² + (coefficient of n) + constant
  • 7 = 2² + 2(coefficient of n) + constant
• Solving the simultaneous equations, we get:
  • Coefficient of n = 1
  • Constant = 0
• Therefore, the nth term is n² + n

Practice and Applications

Practicing identifying the nth term of sequences, both linear and quadratic, is crucial for mastering this concept. Sequences are frequently used in various applications, including:

• Financial mathematics: Calculating compound interest and loan repayments.
• Computer programming: Generating patterns and algorithms.
• Physics: Modelling physical phenomena like projectile motion.

By understanding and applying the concepts of nth term and quadratic sequences, you will gain a strong foundation in algebra and prepare yourself for further mathematical studies.