Sequences and series

Welcome to Sequences and Series!

In this chapter of Pure Mathematics 2 (P2), we are going to explore the world of mathematical patterns. A sequence is simply a list of numbers that follow a specific rule, while a series is what happens when we add those numbers together.

Why does this matter? Well, sequences and series are the "secret code" behind everything from the way plants grow to how your bank calculates interest on savings. Don't worry if it seems like a lot of formulas at first—we will break them down step-by-step!

1. The Basics: What is a Sequence?

A sequence is a set of numbers in a specific order. Each number in the list is called a term. We usually use \(u_n\) or \(x_n\) to represent the "nth term" (the term at position \(n\)).

Finding Terms

There are two main ways to describe a sequence:

• Position-to-term (nth term formula): You are given a formula like \(u_n = 2n + 3\). To find the 1st term, just plug in \(n=1\). To find the 100th term, plug in \(n=100\).
• Recurrence relations: This is like a "stepping stone" rule. You are told how to get to the next term (\(x_{n+1}\)) using the current term (\(x_n\)). For example: \(x_{n+1} = x_n + 5\). If you know the first term, you can find the second, then the third, and so on.

Classifying Sequences

Depending on how they behave, we give sequences special names:

• Increasing: Every term is bigger than the one before it (e.g., 2, 4, 6, 8...).
• Decreasing: Every term is smaller than the one before it (e.g., 10, 7, 4, 1...).
• Periodic: The terms repeat in a cycle. For example: 1, 0, -1, 1, 0, -1... (The order of this sequence would be 3, because it repeats every 3 terms).

Quick Review: Think of a recurrence relation like a recipe. You need the previous ingredient to make the next one. An nth term formula is like a GPS—it takes you straight to any term you want without stopping at the ones in between!

Key Takeaway: Sequences follow rules. If the rule depends on the term before it, it's a recurrence relation. If it depends on the position, it's an nth term formula.

2. Arithmetic Sequences and Series

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

The nth Term Formula

To find any term in an arithmetic sequence, use:
\(u_n = a + (n - 1)d\)
Where:
a = the first term
d = the common difference
n = the position of the term

Arithmetic Series (Summing it up)

When we add the terms of an arithmetic sequence, we get a series. We use \(S_n\) to represent the sum of the first \(n\) terms.
There are two formulas for \(S_n\):
1. Use this if you know the first term (\(a\)) and common difference (\(d\)):
\(S_n = \frac{n}{2}[2a + (n - 1)d]\)

2. Use this if you know the first term (\(a\)) and the very last term (\(l\)):
\(S_n = \frac{n}{2}(a + l)\)

Did you know? (The Gauss Trick)

The story goes that when the famous mathematician Carl Friedrich Gauss was a child, his teacher asked the class to add all numbers from 1 to 100. Gauss did it in seconds! He realized that \(1+100=101\), \(2+99=101\), and so on. He had 50 pairs of 101.
The Syllabus says: You need to know the proof for the sum formula. It involves writing the sum forward and backward, then adding them together—just like Gauss did!

Sigma Notation (\(\sum\))

The symbol \(\sum\) (Sigma) is just a fancy shorthand for "Add them all up."
Example: \(\sum_{r=1}^{5} (2r)\) means: "Plug in 1, 2, 3, 4, and 5 into \(2r\), then add the results."
\(2(1) + 2(2) + 2(3) + 2(4) + 2(5) = 2 + 4 + 6 + 8 + 10 = 30\).

Key Takeaway: Arithmetic = Adding. Use \(u_n = a + (n-1)d\) for terms and the \(S_n\) formulas for totals.

3. Geometric Sequences and Series

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

The nth Term Formula

\(u_n = ar^{n-1}\)
Careful! The power is \((n-1)\), not \(n\). This is a common mistake!

Geometric Series Sum Formula

To find the sum of the first \(n\) terms:
\(S_n = \frac{a(1 - r^n)}{1 - r}\) (This version is easier if \(r < 1\))
OR
\(S_n = \frac{a(r^n - 1)}{r - 1}\) (This version is easier if \(r > 1\))

Using Logs to Find \(n\)

Sometimes a question will ask "How many terms are needed for the sum to exceed 5000?". When you need to find an unknown power (\(n\)), you will usually need to use Logarithms.
Memory Aid: If you have \(r^n = k\), you can "bring the power down" by taking logs: \(n \log(r) = \log(k)\).

Sum to Infinity (\(S_{\infty}\))

Imagine you walk halfway to a wall. Then you walk halfway again. Then halfway again. You will keep moving forever, but you will never actually pass the wall!
In math, if the common ratio is a fraction between -1 and 1 (written as \(|r| < 1\)), the terms get smaller and smaller, approaching zero. We call this a convergent series.
The sum of infinite terms is:
\(S_{\infty} = \frac{a}{1 - r}\)

Key Takeaway: Geometric = Multiplying. If \(|r| < 1\), the series converges to a single number \(S_{\infty}\).

4. The Binomial Expansion

The Binomial Expansion is a shortcut for expanding brackets like \((a + bx)^n\) without having to multiply them manually dozens of times. For P2, we only deal with positive integer values of \(n\).

Tools You Need

• Factorials (!): \(n!\) means multiplying all integers from \(n\) down to 1. E.g., \(4! = 4 \times 3 \times 2 \times 1 = 24\).
• Combinations (\(^nC_r\)): This tells you the coefficients (the numbers in front of the terms). You can find the \(^nC_r\) button on your calculator! It is also written as \(\binom{n}{r}\).

The Formula

\((a + bx)^n = a^n + \binom{n}{1}a^{n-1}(bx) + \binom{n}{2}a^{n-2}(bx)^2 + ... + (bx)^n\)

Step-by-Step Guide:
1. The powers of the first term (\(a\)) start at \(n\) and go down to 0.
2. The powers of the second term (\(bx\)) start at 0 and go up to \(n\).
3. The coefficients come from \(^nC_r\) or Pascal's Triangle.

Common Mistake: When expanding something like \((2 + 3x)^4\), remember to square the whole second term: \((3x)^2\) becomes \(9x^2\), not \(3x^2\)!

Key Takeaway: Binomial expansion is just a pattern. Powers of the first part decrease, powers of the second part increase, and the calculator gives you the "multiplier" (\(^nC_r\)).

Summary Checklist

• Can I find the nth term of arithmetic and geometric sequences?
• Do I know the difference between \(u_n\) (a term) and \(S_n\) (a total)?
• Can I prove the sum formulas for both arithmetic and geometric series?
• Do I know when a geometric series has a sum to infinity? (\(|r| < 1\))
• Can I use \(\sum\) notation?
• Can I expand a binomial expression accurately?

Don't worry if this feels tricky at first! Practice identifying whether a question is Arithmetic (adding) or Geometric (multiplying) before you choose your formula. You've got this!