Introduction to Sequences

Welcome to the study of Sequences! A sequence is simply a list of numbers follow a specific rule or pattern. In Mathematics (Specification B), understanding sequences is about training your eyes to spot these patterns and using algebra to describe them.

Think of a sequence like a set of steps. If you know how high each step is, you can predict exactly how high you will be after ten steps, or even a hundred! Whether you find algebra easy or a bit of a puzzle, this chapter will give you the tools to "crack the code" of any number pattern you encounter.

1. The Basics: What is a Sequence?

A sequence is an ordered list of numbers. Each number in the list is called a term.

  • The first term is the number the sequence starts with.
  • The position of a term (represented by the letter \(n\)) tells us where it is in the list (e.g., \(n=1\) for the first term, \(n=2\) for the second).
  • The rule is the mathematical instruction that tells us how to get from one term to the next or how to find a term based on its position.

Example: Look at the sequence \(3, 7, 11, 15, \dots\)

The first term is \(3\). To get to the next term, we add 4. This "add 4" is the rule for continuing the sequence.

2. Sequences with a Common Difference

One of the most important patterns you need to recognise is a sequence with a common difference. This is often called a linear or arithmetic sequence.

How to spot them:

In these sequences, you add or subtract the same amount every time to get the next number.

Example: \(10, 7, 4, 1, -2, \dots\)

Here, the common difference is \(-3\) because we subtract \(3\) each time.

Continuing the sequence:

To find the next term, simply apply the common difference to the last known term.

Step-by-step: Find the next two terms of \(5, 11, 17, 23, \dots\)

  1. Find the difference: \(11 - 5 = 6\). Check another pair: \(17 - 11 = 6\). The difference is \(+6\).
  2. Add \(6\) to the last term: \(23 + 6 = 29\).
  3. Add \(6\) again: \(29 + 6 = 35\).
  4. The next terms are \(29\) and \(35\).

Quick Tip: If the numbers are going up, the difference is positive. If they are going down, the difference is negative!

3. Common Integer Sequences

The syllabus requires you to recognise special "famous" sequences. You should be able to spot these instantly:

A. Square Numbers

\(1, 4, 9, 16, 25, 36, \dots\)

The Rule: These are the results of \(n^2\) (where \(n = 1, 2, 3, \dots\)). For example, the 4th term is \(4 \times 4 = 16\).

B. Cube Numbers

\(1, 8, 27, 64, 125, \dots\)

The Rule: These are the results of \(n^3\). For example, the 3rd term is \(3 \times 3 \times 3 = 27\).

C. Prime Numbers

\(2, 3, 5, 7, 11, 13, \dots\)

The Rule: Numbers that have exactly two factors: \(1\) and themselves. Note: \(1\) is not a prime number!

D. Fibonacci-style Sequences

\(1, 1, 2, 3, 5, 8, 13, \dots\)

The Rule: You find the next term by adding the two previous terms together. For example, \(5 + 8 = 13\).

4. Finding the \(n^{th}\) Term (Linear)

Sometimes, a question might ask you for a general rule for the \(n^{th}\) term of a sequence with a common difference. This allows you to find any term (like the 100th) without writing the whole list out!

The formula for a linear sequence is always written in the form:
\(dn + c\)

  • \(d\) is the common difference.
  • \(c\) is the "zero term" (the value the sequence would have at position \(n=0\)).

Step-by-Step Example: Find the rule for \(5, 8, 11, 14, \dots\)

  1. Find \(d\): The sequence goes up by \(3\) each time. So, \(d = 3\). Our rule starts with \(3n\).
  2. Find \(c\): Ask yourself, "What would come before the first term?" If the first term is \(5\), and we subtract our difference (\(3\)), we get \(5 - 3 = 2\). So, \(c = 2\).
  3. Put it together: The \(n^{th}\) term rule is \(3n + 2\).

Check your work: To see if it works for the 4th term (\(n=4\)): \(3(4) + 2 = 12 + 2 = 14\). It matches the sequence!

5. Common Mistakes to Avoid

  • Confusing \(n\) with the term value: Remember that \(n\) is the position (\(1^{st}, 2^{nd}, 3^{rd}\)), not the actual number in the list.
  • Getting the sign wrong: If a sequence is decreasing (e.g., \(10, 8, 6 \dots\)), the difference is negative (\(-2\)). Your \(n^{th}\) term must start with \(-2n\).
  • Mistaking Prime Numbers: Many students think \(9\) or \(15\) are prime because they are odd. Always check if you can divide the number by anything other than \(1\) and itself!

Key Takeaways

Summary Box:

  • Linear Sequences: Have a common difference. Use \(dn + c\) to find the rule.
  • To continue a sequence: Identify the pattern (addition, subtraction, or a special sequence) and apply it to the last term.
  • Recognise Squares: \(1, 4, 9, 16, 25\dots\)
  • Recognise Cubes: \(1, 8, 27, 64, 125\dots\)
  • Recognise Primes: \(2, 3, 5, 7, 11\dots\)

Don't worry if this seems tricky at first! The more sequences you look at, the faster your brain will become at spotting the "hidden" rules. Keep practicing!

Cross-reference: For more complex algebraic rules involving brackets or indices, see the chapter on Algebraic manipulation and formulae.