Sequences

Welcome to the World of Patterns!

In this chapter, we are going to explore Sequences. Simply put, a sequence is just a list of numbers that follows a specific rule. Whether you are looking at the growth of a population, the interest in a bank account, or even the patterns in a flower, sequences are everywhere!

Don’t worry if you find some of the formulas a bit intimidating at first. We will break them down step-by-step, using clear examples and analogies to make sure you feel confident. Let’s dive in!

1. The Basics: What is a Sequence?

A sequence is a list of numbers ordered by "position." We usually write the terms as \(u_1, u_2, u_3, ...\) where the small number (the subscript) tells you the position of the term.

Sequence vs. Series

Students often mix these up, but here is a simple way to remember:
• A Sequence is the list of numbers: \(2, 4, 6, 8\).
• A Series is the sum of those numbers: \(2 + 4 + 6 + 8\).

Finite vs. Infinite

• Finite sequences have a specific end (like the countdown to a rocket launch).
• Infinite sequences go on forever (like the set of all even numbers).

Quick Review:
\(n\) = the position of the term (always a whole number like 1, 2, 3...).
\(u_n\) = the actual value of the term at that position.

Key Takeaway: A sequence is a list, and a series is a sum. Use \(n\) for the position and \(u_n\) for the value.

2. Generating Sequences

There are two main ways to describe a sequence's rule:

The \(n^{th}\) Term Formula (Deductive)

This is like a "position-to-term" machine. If you know the position \(n\), you can calculate the value \(u_n\) directly.
Example: If \(u_n = 3n + 1\):
To find the 1st term (\(n=1\)): \(3(1) + 1 = 4\).
To find the 100th term (\(n=100\)): \(3(100) + 1 = 301\).

Recurrence Relations (Inductive)

This is a "term-to-term" rule. It tells you how to get the next term from the current one. It usually looks like \(u_{n+1} = f(u_n)\).
Example: \(u_{n+1} = u_n + 5\), with \(u_1 = 2\).
This means "to get the next term, add 5 to the current one."
Terms: \(2, 7, 12, 17, ...\)

Did you know? Recurrence relations are like a relay race—each runner (term) passes the baton to the next one based on the rule.

Key Takeaway: Use the \(n^{th}\) term formula to jump to any position; use recurrence relations to build the sequence step-by-step.

3. Mathematical Behavior

We can describe how a sequence "behaves" as it moves along:

• Increasing: Every term is larger than the one before it (\(u_{n+1} > u_n\)).
• Decreasing: Every term is smaller than the one before it (\(u_{n+1} < u_n\)).
• Periodic: The terms repeat in a cycle.
Example: \(1, 0, -1, 1, 0, -1, ...\) has a period of 3 because it repeats every 3 terms.

Common Mistake: Thinking a sequence is periodic just because it has similar numbers. It must repeat exactly and in the same order!

Key Takeaway: Look at the relationship between \(u_n\) and \(u_{n+1}\) to decide if it’s increasing, decreasing, or periodic.

4. Arithmetic Progressions (AP)

An Arithmetic Progression is a sequence where you add (or subtract) the same amount every time. This amount is called the common difference (\(d\)).

The Formulas

• First term = \(a\)
• Common difference = \(d\)
• \(n^{th}\) term: \(u_n = a + (n - 1)d\)
• Sum of \(n\) terms (\(S_n\)):
\(S_n = \frac{n}{2}(2a + (n - 1)d)\)
OR
\(S_n = \frac{n}{2}(a + l)\) where \(l\) is the last term.

Memory Aid: For the \(n^{th}\) term, why is it \((n-1)\)? Because you don't add the difference to the first term. To get to the 3rd term, you only take 2 "steps" of \(d\).

Step-by-Step Example:
Find the sum of the first 10 terms of \(5, 8, 11, ...\)
1. Identify \(a = 5\) and \(d = 3\).
2. We want \(S_{10}\), so \(n = 10\).
3. Plug into formula: \(S_{10} = \frac{10}{2}(2(5) + (10 - 1)3)\)
4. Calculate: \(5(10 + 27) = 5(37) = 185\).

Key Takeaway: Arithmetic sequences are about adding. Always find \(a\) and \(d\) first!

5. Geometric Progressions (GP)

A Geometric Progression is a sequence where you multiply by the same amount every time. This is called the common ratio (\(r\)).

The Formulas

• First term = \(a\)
• Common ratio = \(r\)
• \(n^{th}\) term: \(u_n = ar^{n-1}\)
• Sum of \(n\) terms (\(S_n\)): \(S_n = \frac{a(1 - r^n)}{1 - r}\) (or \(\frac{a(r^n - 1)}{r - 1}\))

Important Note: If \(r\) is between -1 and 1 (\(|r| < 1\)), the sequence is convergent. This means the terms get smaller and smaller, heading towards zero. If \(|r| \ge 1\), it is divergent.

Analogy: Imagine a bouncing ball that reaches half its previous height each time. The heights form a GP where \(r = 0.5\).

Key Takeaway: Geometric sequences are about multiplying. Find \(a\) and \(r\) to solve most problems.

6. Sum to Infinity (\(S_\infty\))

If a geometric series is convergent (\(|r| < 1\)), we can actually find the sum of all the terms, even if they go on forever! This is because the terms eventually become so tiny they don't add anything meaningful to the total.

The Formula:
\(S_\infty = \frac{a}{1 - r}\)

Don't worry if this seems tricky: It feels weird that you can add up infinite numbers and get a finite answer. Just remember the "Halfway to the Wall" analogy: If you keep walking half the distance to a wall, you will never technically cross it, but your total distance will eventually just be the distance to the wall!

Key Takeaway: You can only find \(S_\infty\) if \(-1 < r < 1\). If \(r\) is 2, the sum just keeps growing to infinity!

7. Sigma Notation (\(\Sigma\))

The symbol \(\Sigma\) (Sigma) is just a fancy Greek "S" that stands for Sum. It’s a shorthand way of writing a series.

\(\sum_{r=1}^{n} u_r\)

• The number at the bottom is where you start (usually \(r=1\)).
• The number at the top is where you stop.
• The part in the middle is the rule for each term.

Quick Review: To expand a Sigma sum, just plug in \(r=1\), then \(r=2\), then \(r=3\)... all the way to the top number, and add them all up.

Key Takeaway: Sigma notation is just a set of instructions: "Start here, end there, follow this rule, and add them up."

8. Modelling with Sequences

Sequences are very useful for real-life math:

• Simple Interest: This usually forms an Arithmetic Progression because you add the same amount of interest every year.
• Compound Interest: This forms a Geometric Progression because you multiply the balance by an interest factor (like \(1.05\)) every year.
• Growth and Decay: Bacteria populations or radioactive decay are often modeled using GPs.

Common Mistake: When solving growth problems (like "when will the population exceed 5000?"), you will often end up with an inequality like \(2^n > 100\). You will need to use Logarithms to solve for \(n\). Always remember that \(n\) must be a whole number, so round up if necessary!

Key Takeaway: Identify if the change is "adding a fixed amount" (AP) or "multiplying by a percentage" (GP).

Final Checklist for Success

• Can you identify if a sequence is Arithmetic or Geometric?
• Do you know the difference between \(u_n\) (the term) and \(S_n\) (the sum)?
• Are you comfortable using Logarithms to find \(n\) in a GP?
• Remember: For any AP, you need \(a\) and \(d\). For any GP, you need \(a\) and \(r\)!

You've got this! Practice identifying the "a", "d", and "r" values first, and the rest is just using the formulas.