Welcome to the World of Sequences and Series!

In this chapter of Additional Pure Mathematics, we are going to dive deep into the patterns that govern numbers. Think of a sequence as a digital "recipe"—if you know the rules, you can predict what happens next, whether it’s in a simple number line or a complex model of a biological population. We will explore how sequences behave in the long run and learn the powerful tools needed to solve "recurrence relations," which are the discrete cousins of the differential equations you’ve met in Core Pure!

1. The Basics: Recurrence vs. Position-to-Term

A sequence is simply a list of numbers in a specific order, usually written as \( \{u_n\} \). You might start your list at \( u_1 \), but in Further Maths, we often start with a zeroth term, \( u_0 \). Don't let that throw you; it's just the starting point!

Two Ways to Define a Sequence:

  • Recurrence Relations: These tell you how to get to the next term using the current one.
    Example: \( u_{n+1} = 2u_n + 3 \). To find the next number, you double the current one and add three. It’s like a breadcrumb trail—you need the previous step to find the next one.
  • Position-to-Term (Closed Form): This is a direct formula \( u_n = f(n) \).
    Example: \( u_n = 3n + 1 \). If you want the 100th term, you just plug in \( n = 100 \). This is like having a GPS coordinate—you can jump straight to any point!

Did you know? Computers love recurrence relations because they are "iterative"—the computer just performs the same simple calculation over and over again to generate complex patterns.

Key Takeaway: A recurrence relation defines a step-by-step rule, while a closed-form formula gives you the answer for any \( n \) immediately.

2. Describing how Sequences Behave

When we look at a sequence as \( n \) gets very large (\( n \to \infty \)), we use specific terms to describe what we see:

  • Convergence: The terms get closer and closer to a single fixed value (a limit). We call this a steady-state.
  • Divergence: The terms do not settle down. They might head off to infinity (\( \infty \)) or minus infinity (\( -\infty \)), or just wander around.
  • Periodic: The sequence repeats itself in a cycle. A sequence like \( 1, 2, 1, 2, ... \) has a period of 2.
  • Oscillating: The terms swing back and forth, often on either side of a central value. Note that an oscillating sequence can be convergent (getting closer to a value) or divergent (swinging wider and wider).
  • Monotonic: The sequence only moves in one direction—either always increasing or always decreasing.

Quick Review Box:
- Bounded: A sequence that stays within a certain "ceiling" and "floor" (e.g., it never goes above 10 or below -10).
- Unbounded: A sequence that eventually escapes any limit you set.

3. Solving First-Order Recurrence Relations

A first-order linear recurrence relation looks like this: \( u_{n+1} = au_n + f(n) \).
To solve these (finding the "closed form"), we use a two-part strategy exactly like solving differential equations!

The Step-by-Step Method:

  1. Find the Complementary Function (CF): Solve the "homogenous" part where \( f(n) = 0 \). For \( u_{n+1} = au_n \), the solution is always \( u_n = A(a)^n \).
  2. Find the Particular Integral (PI): Look at \( f(n) \) and guess a form for \( u_n \):
    • If \( f(n) \) is a constant, try \( u_n = \lambda \).
    • If \( f(n) \) is a polynomial (like \( 3n + 2 \)), try \( u_n = \lambda n + \mu \).
    • If \( f(n) \) is an exponential (like \( 5^n \)), try \( u_n = \lambda(5^n) \).
  3. General Solution: Add them together! \( u_n = \text{CF} + \text{PI} \).
  4. Find the Constant: Use the initial condition (like \( u_0 \)) to find the value of \( A \).

Common Mistake: If your "guess" for the PI is already part of your CF, you must multiply your guess by \( n \) to make it work!

4. Solving Second-Order Recurrence Relations

These involve the two previous terms: \( u_{n+2} + au_{n+1} + bu_n = f(n) \). Don't worry if this looks scary; the process is very logical.

The Auxiliary Equation:

We start by looking at the homogeneous version and creating an auxiliary equation: \( m^2 + am + b = 0 \). The roots of this equation tell us the shape of our solution:

  • Case 1: Two Distinct Real Roots (\( \alpha, \beta \))
    The CF is \( u_n = A\alpha^n + B\beta^n \).
  • Case 2: Repeated Real Roots (\( \alpha \))
    The CF is \( u_n = (A + Bn)\alpha^n \).
  • Case 3: Complex Roots (\( p \pm iq \))
    The CF will involve powers of the modulus and arguments (using \( r^n \cos(n\theta) \) and \( r^n \sin(n\theta) \) forms).

Memory Aid: This is the "Discrete Analogue." If you can solve \( y'' + ay' + by = 0 \), you can solve these! The only difference is we use \( \alpha^n \) instead of \( e^{kx} \).

5. Fibonacci and Lucas Numbers

The most famous second-order sequence is the Fibonacci Sequence: \( u_{n+2} = u_{n+1} + u_n \), with \( u_0 = 0, u_1 = 1 \).
The Lucas Sequence uses the same rule but starts with \( u_0 = 2, u_1 = 1 \).

The Golden Ratio (\( \phi \)):

The auxiliary equation for Fibonacci is \( m^2 - m - 1 = 0 \). The positive root is \( \phi = \frac{1+\sqrt{5}}{2} \approx 1.618 \).
Did you know? As \( n \to \infty \), the ratio of successive terms \( \frac{u_{n+1}}{u_n} \) for any Fibonacci-like sequence converges to \( \phi \). You see this everywhere in nature, from pinecones to galaxies!

6. Proof by Induction

The syllabus requires you to prove results for sequences and series using Mathematical Induction. It’s like a ladder: if you can get on the first rung and prove that each rung leads to the next, you can reach the top!

The Four Steps of Power:

  1. Basis: Prove the statement is true for the first term (usually \( n=0 \) or \( n=1 \)).
  2. Assumption: Assume the statement is true for \( n=k \).
  3. Inductive Step: Use your assumption to prove it must be true for \( n=k+1 \).
  4. Conclusion: Write the formal sentence: "Since it is true for \( n=1 \), and if true for \( n=k \) it is true for \( n=k+1 \), then by induction it is true for all \( n \in \mathbb{Z}^+ \)."

Key Takeaway: Never skip the conclusion! It's where you pick up easy "communication" marks in the exam.

7. Modelling and the INT Function

In the real world, populations of animals or interest in bank accounts change in "jumps." We use recurrence relations to model this.

Sometimes we use the \( INT(x) \) function (or "floor" function). It rounds a number down to the nearest integer.
Example: If a population model predicts \( 10.7 \) rabbits, \( INT(10.7) = 10 \). You can't have \( 0.7 \) of a rabbit!
Expect to see models involving birth rates (adding a percentage) and death rates or harvesting (subtracting a constant).

Encouraging Phrase: Don't worry if the second-order non-homogeneous equations feel long—they are just a series of small, logical steps. Practice identifying the "form" of the Particular Integral, and the rest is just algebra!

Summary Checklist

  • Can you identify if a sequence is convergent, periodic, or oscillating?
  • Do you know the 3 cases for the roots of a second-order auxiliary equation?
  • Can you perform a proof by induction for a summation series?
  • Do you remember that the CF for \( u_{n+1} = au_n \) is always \( A(a)^n \)?