Welcome to the World of Sequences and Series!

In this chapter of Additional Pure Mathematics, we are going to explore the patterns that govern numbers. Think of a sequence as a digital heartbeat or the way a population grows over time. We will learn how to describe these patterns, predict where they are going, and even find the secret mathematical "DNA" that creates them. Don’t worry if this seems a bit abstract at first—we’ll break it down step-by-step!

1. Defining Sequences: Two Ways to Tell a Story

A sequence is just a list of numbers following a specific rule. In Further Maths, we use the notation \(\{u_n\}\) to represent a sequence. There are two main ways to define them:

Recurrence Relations

A recurrence relation tells you how to get to the next term using the current one. It’s like a set of instructions: "To get tomorrow’s value, take today’s value and double it."
Example: \(u_{n+1} = 2u_n + 3\), with a starting value \(u_0 = 1\).

Position-to-Term (Closed Form)

A closed form formula allows you to calculate any term directly without knowing the one before it. It’s like having a GPS coordinate for every point in the sequence.
Example: \(u_n = 3n + 5\). To find the 100th term, just plug in \(n=100\).

Quick Review: Recurrence relations need a starting point (like \(u_0\) or \(u_1\)) to get moving!

2. Describing Sequence Behaviour

How does the sequence behave as it goes on forever? We use specific "character traits" to describe them:

  • Convergence: The terms get closer and closer to a specific number (a limit).
  • Divergence: The terms head off to infinity (or negative infinity) or fail to settle down.
  • Periodic: The sequence repeats itself in a cycle (e.g., \(1, 2, 1, 2, ...\)). A period of two is often called oscillating.
  • Monotonic: The sequence only ever goes up, or only ever goes down. It never changes direction.

Did you know? A sequence can be both oscillating and convergent! Imagine a pendulum swinging: the swings get smaller and smaller (converging to zero) while moving back and forth (oscillating).

Key Takeaway: We look at the limit as \(n \to \infty\) to find the steady-state. If \(u_{n+1} = u_n = L\) at the limit, we can solve for \(L\).

3. Solving First-Order Linear Recurrence Relations

This is a big part of the syllabus. We want to turn a recurrence relation into a closed form. The general form is \(u_{n+1} = au_n + f(n)\).

Step 1: The Complementary Function (CF)

First, look at the "homogeneous" part: \(u_{n+1} - au_n = 0\).
The solution is always in the form: \(u_n = A(a)^n\).

Step 2: The Particular Solution (PS)

Now we look at \(f(n)\). We "guess" a form for the PS based on what \(f(n)\) looks like:

  • If \(f(n)\) is a constant (like \(5\)), try \(u_n = \lambda\).
  • If \(f(n)\) is linear (like \(3n + 2\)), try \(u_n = \lambda n + \mu\).
  • If \(f(n)\) is an exponential (like \(k^n\)), try \(u_n = \lambda k^n\).

Step 3: The General Solution

The total answer is: General Solution = CF + PS.
Finally, use your initial condition (like \(u_0\)) to find the value of the constant \(A\).

Common Mistake: Forgetting to add the CF and PS before using the initial condition to find \(A\). Always combine them first!

4. Fibonacci and the Golden Ratio

The Fibonacci sequence (\(1, 1, 2, 3, 5, 8, ...\)) is defined by \(u_{n+1} = u_n + u_{n-1}\). Closely related are Lucas numbers, which use the same rule but start with \(2, 1\).

The star of this section is the Golden Ratio (\(\phi\)).
\(\phi = \frac{1 + \sqrt{5}}{2} \approx 1.618\)

Important Properties of \(\phi\):
1. \(\phi^2 = \phi + 1\)
2. \(\frac{1}{\phi} = \phi - 1\)
3. As \(n\) gets larger, the ratio of successive Fibonacci terms \(\frac{u_{n+1}}{u_n}\) converges to \(\phi\).

Analogy: Think of \(\phi\) as the "growth factor" of nature. You see it in pinecones, sunflowers, and even galaxies!

5. Proof by Induction with Sequences

You may be asked to prove that a closed-form formula is correct for a given recurrence relation. We use Mathematical Induction:

  1. Base Case: Show it works for \(n = 0\) (or \(n = 1\)).
  2. Assumption: Assume the formula is true for \(n = k\).
  3. Inductive Step: Use the recurrence relation and your assumption to show it must then be true for \(n = k+1\).
  4. Conclusion: State that since it works for the base case and the step holds, it is true for all \(n\).

Memory Aid: Induction is like a row of dominoes. The Base Case is knocking over the first domino. The Inductive Step proves that if one falls, the next one must fall. Together, they all go down!

6. Modelling: Real-World Maths

Recurrence relations are perfect for modelling things like population growth or loan repayments.
Example: A population of rabbits increases by 10% each year, but 50 rabbits are removed.
This is modelled by: \(u_{n+1} = 1.1u_n - 50\).

In these problems, you might use the INT(x) function to keep your numbers as whole integers (since you can't have half a rabbit!).

Key Takeaway: Modelling is just turning a word problem into a recurrence relation so you can solve it using the steps we learned in Section 3.

Summary: Your "Quick Review" List

  • Notation: \(\{u_n\}\) for sequences.
  • Behaviour: Convergent, Divergent, Periodic, Monotonic.
  • Limits: Find them by setting \(u_{n+1} = u_n = L\).
  • Solving: Use the CF + PS method for first-order relations.
  • Golden Ratio: Know \(\phi = \frac{1+\sqrt{5}}{2}\) and its link to Fibonacci.
  • Proof: Use the standard 4-step Induction process.