🚀 Welcome to Sequences: Finding Patterns in Math!

Hello everyone! Don't worry if numbers sometimes feel random; in this chapter, we learn how to find the hidden rules and patterns that govern groups of numbers. This skill, called Sequences, is fundamental to advanced mathematics and helps us predict the future (mathematically speaking!).

We will learn how to:

  • Identify different types of sequences.
  • Generate terms using a rule.
  • Find the formula (the n-th term) for any sequence.

1. The Basics: What is a Sequence?

A Simple Definition

A Sequence is just an ordered list of numbers. The order matters!

Example:
\(2, 5, 8, 11, 14, ...\)

Key Terminology

Each number in the sequence is called a Term. We use special notation to identify the position of a term:

  • The 1st term is called \(T_1\). (In the example, \(T_1 = 2\))
  • The 2nd term is called \(T_2\). (In the example, \(T_2 = 5\))
  • The \(n\)-th term is called \(T_n\). This is the formula we use to find any term's value, provided we know its position (\(n\)).

💡 Quick Review: \(n\) is the position (1st, 2nd, 3rd, etc.), and \(T_n\) is the value of the number in that position.

2. Linear Sequences (Arithmetic Progressions)

What makes a sequence Linear?

A sequence is Linear (or Arithmetic) if you add or subtract the exact same number every time to get from one term to the next.

This constant number is called the Common Difference (d).

Example: \(10, 15, 20, 25, ...\)
Here, \(d = +5\).

Example: \(50, 48, 46, 44, ...\)
Here, \(d = -2\).

Finding the \(n\)-th Term for a Linear Sequence

Finding the general rule (\(T_n\)) is the most important skill here. This rule lets you jump straight to the 100th term without listing all 99 before it!

The general form for a linear sequence is:
\[T_n = dn + c\]
where \(d\) is the common difference, and \(c\) is a constant adjustment.

Step-by-Step Guide: The "Zero Term" Method

Let's find the \(n\)-th term for the sequence: \(4, 7, 10, 13, ...\)

  1. Find the Common Difference (\(d\)):
    \(7 - 4 = 3\). \(d = 3\).

    Since the difference is 3, the rule must involve \(3n\).

  2. Check the \(3n\) table:
    If the sequence was just \(3n\), the terms would be: \(3 \times 1 = 3\), \(3 \times 2 = 6\), \(3 \times 3 = 9\), ...

    The table is: \(3, 6, 9, 12, ...\)

  3. Find the Adjustment (\(c\)):
    Compare your actual sequence (\(4, 7, 10, ...\)) with the \(3n\) sequence (\(3, 6, 9, ...\)).
    How do you get from 3 to 4? You add 1.
    How do you get from 6 to 7? You add 1.
    The adjustment is \(+1\).
  4. Write the final formula:
    \[T_n = 3n + 1\]

⭐ Memory Trick: The Zero Term ⭐
The adjustment \(c\) is always the number that comes before the first term (\(T_1\)). We call this the Zero Term (\(T_0\)).
Sequence: \((1), 4, 7, 10, 13, ...\) (The difference is +3, so \(4 - 3 = 1\)).
The zero term is 1. Thus, \(T_n = 3n + 1\).

Key Takeaway: Linear sequences are built from multiplication (the difference) and addition/subtraction (the zero term).

3. Generating Terms from a Formula

Sometimes, you are given the \(n\)-th term formula and asked to list the first few terms. This is much easier!

Process: Substitution is Key

Find the first three terms of the sequence given by the rule: \(T_n = 5n - 2\).

  1. For the 1st term (\(n=1\)):
    \(T_1 = 5(1) - 2 = 5 - 2 = 3\)
  2. For the 2nd term (\(n=2\)):
    \(T_2 = 5(2) - 2 = 10 - 2 = 8\)
  3. For the 3rd term (\(n=3\)):
    \(T_3 = 5(3) - 2 = 15 - 2 = 13\)

The sequence is: \(3, 8, 13, ...\)

🛑 Common Mistake to Avoid:
Students sometimes substitute the term value (\(T_n\)) into the formula instead of the position (\(n\)). Remember: \(n\) is always 1, 2, 3, 4, ...

4. Quadratic Sequences (The Double Difference)

Not all sequences have a constant difference on the first try. If the difference changes steadily, the sequence is probably Quadratic. This means the formula will contain an \(n^2\) term.

Example: The sequence of square numbers: \(1, 4, 9, 16, 25, ...\) (The rule is simply \(T_n = n^2\)).

Finding the \(n\)-th Term for a Quadratic Sequence

This takes a little more work, but the method is always the same. We use the Method of Differences.

Let's find the rule for the sequence: \(5, 9, 15, 23, 33, ...\)

  1. Find the First Difference: (Subtract the previous term from the current term)
    \(5 \quad 9 \quad 15 \quad 23 \quad 33\)
    +4 \quad +6 \quad +8 \quad +10

    The first difference is NOT constant, so it's not linear.

  2. Find the Second Difference: (Find the difference between the differences)
    4 \quad 6 \quad 8 \quad 10
    +2 \quad +2 \quad +2

    The Second Difference is constant (+2)! This confirms it is a quadratic sequence.

  3. Determine the \(n^2\) part:
    The coefficient of \(n^2\) is always half the second difference.
    Second Difference = 2. Half of 2 is 1.
    So, the formula starts with \(1n^2\) (or simply \(n^2\)).
  4. Subtract the \(n^2\) component:

    This is like peeling the quadratic layer off to reveal the remaining linear sequence.

    Position (\(n\)) 1 2 3 4
    Original Sequence (\(T_n\)) 5 9 15 23
    \(n^2\) (or \(1n^2\)) \(1^2 = 1\) \(2^2 = 4\) \(3^2 = 9\) \(4^2 = 16\)
    Remaining Sequence (\(T_n - n^2\)) \(5 - 1 = 4\) \(9 - 4 = 5\) \(15 - 9 = 6\) \(23 - 16 = 7\)
  5. Find the rule for the Remaining Sequence:
    The remaining sequence is: \(4, 5, 6, 7, ...\)
    This is a simple linear sequence!
    Difference \(d = +1\). Zero Term (\(4 - 1\)) = 3.
    Rule for remaining part: \(1n + 3\) (or \(n + 3\)).
  6. Combine the parts:
    \[T_n = (\text{Quadratic Part}) + (\text{Linear Part})\] \[T_n = n^2 + n + 3\]

Key Takeaway: If the second difference is \(2a\), the formula starts with \(an^2\).

5. Other Important Sequence Types

5a. Geometric Sequences

A Geometric Sequence is created by multiplying (or dividing) by the same fixed number each time. This number is called the Common Ratio (r).

At this level, you mainly need to be able to identify them and generate terms.

Example 1: \(3, 6, 12, 24, 48, ...\) (Ratio \(r=2\). We multiply by 2.)

Example 2: \(100, 50, 25, 12.5, ...\) (Ratio \(r=0.5\) or \(\frac{1}{2}\). We multiply by \(\frac{1}{2}\).)

5b. The Fibonacci Sequence (A Recurrence Relation)

This is a famous sequence that shows up frequently in nature (like sunflower seed patterns!). It is special because the rule depends on the previous terms, not just the position \(n\).

The rule for the Fibonacci Sequence is: Add the two previous terms to get the next term.

It usually starts: \(1, 1, ...\)

\[T_n = T_{n-1} + T_{n-2}\]

The sequence is: \(1, 1, 2, 3, 5, 8, 13, 21, ...\)

Did you know? The ratio between consecutive Fibonacci numbers gets closer and closer to the famous Golden Ratio (\(\phi\)), which is approximately 1.618.

📚 Chapter Review: Key Concepts

Quick Check List:
  • Linear (Arithmetic): Constant first difference (\(d\)). Rule is \(T_n = dn + c\).
  • Quadratic: Constant second difference (\(2a\)). Rule contains \(an^2\).
  • Geometric: Constant ratio (\(r\)). Uses multiplication/division.
  • Generating Terms: Substitute \(n = 1, 2, 3, ...\) into the formula.

You've made great progress! Finding patterns is the key to mastering this topic. Practice your "double difference" method for quadratic sequences until it feels automatic. You've got this!