Welcome to Sequences and the \( n \)th Term

In this chapter of the Algebra section, we are going to explore the world of number patterns. A sequence is simply a list of numbers that follow a specific rule. Understanding these patterns is like learning a secret code—once you know the rule, you can predict any number in the sequence, even if it's the 1,000th one! Whether you are looking at simple house numbers or complex computer algorithms, sequences are everywhere.

1. Understanding the Language of Sequences

Before we dive into the calculations, let's look at the two main ways we describe a sequence:

Term-to-term rule: This tells you how to get from one number to the next. For example, in the sequence \( 5, 8, 11, 14... \), the rule is "add 3."

Position-to-term rule (\( n \)th term): This is much more powerful. It connects the position of the number (is it 1st, 2nd, or 100th?) to the value of the number. We use the letter \( n \) to represent the position.

Quick Review:
If \( n = 1 \), we are talking about the 1st term.
If \( n = 10 \), we are talking about the 10th term.
If the rule is \( 2n \), and we want the 5th term, we just calculate \( 2 \times 5 = 10 \).

2. Special Number Sequences

There are some sequences that appear so often in GCSE Maths that you should be able to recognise them instantly:

  • Square Numbers: \( 1, 4, 9, 16, 25, ... \) (The rule is \( n^2 \))
  • Cube Numbers: \( 1, 8, 27, 64, 125, ... \) (The rule is \( n^3 \))
  • Triangular Numbers: \( 1, 3, 6, 10, 15, ... \) (Think of these as dots forming a triangle!)
  • Fibonacci-type Sequences: In these sequences, you find the next term by adding the two terms before it. The most famous one is \( 1, 1, 2, 3, 5, 8, 13, ... \) (because \( 1+1=2, 1+2=3, 2+3=5 \), and so on).

3. Arithmetic Progressions (Linear Sequences)

An Arithmetic Progression is a sequence where the difference between terms is always the same. We often call these Linear Sequences because they grow at a steady rate, just like a straight-line graph.

How to find the \( n \)th term of a Linear Sequence

Don't worry if this seems tricky; just follow these three simple steps. Let’s find the \( n \)th term for: \( 7, 12, 17, 22, ... \)

  1. Find the common difference: How much is it changing by each time? Here, it is \( +5 \). This means our rule starts with \( 5n \).
  2. Write out the \( 5n \) table: Under your sequence, write the 5 times table: \( 5, 10, 15, 20 \).
  3. Work out the adjustment: How do you get from your 5 times table to your actual sequence? From \( 5 \) to \( 7 \), you add \( 2 \). From \( 10 \) to \( 12 \), you add \( 2 \).

The Final Rule: \( 5n + 2 \)

Common Mistake to Avoid:

Sometimes the sequence goes down, e.g., \( 10, 7, 4, 1, ... \). The difference here is \( -3 \), so your rule must start with \( -3n \). Don't forget the minus sign!

4. Geometric Progressions

In a Geometric Progression, we don't add or subtract to get the next term; we multiply (or divide) by a fixed number called the common ratio (\( r \)).

The rule usually looks like \( r^n \). For example, in the sequence \( 3, 9, 27, 81, ... \), each number is being multiplied by \( 3 \). The common ratio is \( 3 \).

Note for Higher Tier: The common ratio \( r \) can be a fraction, a decimal, or even a surd (like \( \sqrt{2} \)).

5. Quadratic Sequences (Higher Tier Only)

If the difference between the terms is not the same, it might be a Quadratic Sequence. You can tell it's quadratic if the second row of differences is constant.

Example: \( 2, 6, 12, 20, 30... \)

First differences: \( 4, 6, 8, 10 \)
Second differences: \( 2, 2, 2 \)

Because the second difference is constant, the sequence involves \( n^2 \).

How to find the \( n \)th term for Quadratic Sequences:

  1. Find the second difference.
  2. Halve it to find the coefficient of \( n^2 \). If the second difference is \( 2 \), the sequence starts with \( 1n^2 \).
  3. Subtract the values of \( n^2 \) from your original sequence to find the "remainder."
  4. The remainder will be a simple linear sequence. Find the \( n \)th term for that and add it to your \( n^2 \) part.

6. Summary and Key Takeaways

Key Terms:
\( n \): The position of the term (1st, 2nd, 3rd...).
Linear Sequence: Changes by adding/subtracting the same amount.
Geometric Sequence: Changes by multiplying/dividing by the same amount.
\( n \)th term: An algebraic formula that works for any position in the sequence.

Quick Tip for the Exam:
If a question asks you to "find the 10th term" and you already have the \( n \)th term formula, just replace \( n \) with \( 10 \) and solve. If you don't have the formula and the sequence is short, you can just keep adding the difference until you reach the 10th number!

Did you know? The Fibonacci sequence is found throughout nature, from the petals on a flower to the shape of spiral galaxies!