Introduction to Sequences and Series
Welcome! In this chapter, we are going to explore the world of mathematical patterns. A sequence is simply a list of numbers that follows a specific rule, while a series is what we get when we add those numbers together. These concepts are incredibly important because they allow us to model everything from how interest grows in a bank account to how many people might catch a flu virus over time. Don't worry if it seems like a lot of formulas at first—once you see the patterns, it all starts to click!
1. Binomial Expansion (Section 4.1)
The Binomial Expansion is a way of "opening up" brackets like \((a + b)^n\) without having to multiply them manually over and over again.
Positive Integer Powers
When \(n\) is a whole positive number (like 2, 3, or 10), we use the Binomial Theorem. You might remember Pascal’s Triangle from earlier studies; the numbers in the triangle provide the coefficients (the numbers in front of the terms).
Key terms to know:
n! (n factorial): This means multiplying a number by every whole number below it down to 1. For example, \(4! = 4 \times 3 \times 2 \times 1 = 24\).
\(^nC_r\) or \(\binom{n}{r}\): This is the formula for finding a specific coefficient. You can find this button on your calculator!
Expansion for Any Rational Power
When \(n\) is a fraction or a negative number, the expansion doesn't stop—it goes on forever! For these, we usually use the formula for \((1 + x)^n\):
\( (1 + x)^n = 1 + nx + \frac{n(n-1)}{2!}x^2 + \frac{n(n-1)(n-2)}{3!}x^3 + ... \)
Important Note: Range of Validity
Because this expansion goes on forever, it only "works" (converges) if the value of \(x\) is small. Specifically, for the expansion of \((1 + bx)^n\), it is only valid when \(|bx| < 1\). If you are expanding \((a + bx)^n\), you must factor out the \(a\) first to get a "1" inside the bracket.
Example: For \((1 + 2x)^{-1}\), the expansion is valid when \(|2x| < 1\), which means \(|x| < 0.5\).
Key Takeaway:
Always check if your expansion is valid. If the first number in the bracket isn't 1, factor it out before you start expanding!
2. Working with Sequences (Section 4.2)
A sequence is just a string of numbers. We use \(u_n\) to represent the "\(n\)th term" (the value of the number at position \(n\)).
Types of Sequences
- Increasing: Every term is bigger 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 sequence repeats itself in a cycle. For example, \(3, 5, 3, 5, 3, 5...\) is periodic with an "order" of 2.
Recurrence Relations
Sometimes, a sequence is defined by how it relates to the term before it. This is written as \(u_{n+1} = f(u_n)\).
Analogy: Think of this like a "Follow the Leader" game. To know what the next person does, you have to look at what the current person is doing.
Quick Review:
If \(u_{n+1} = u_n + 3\) and \(u_1 = 5\):
\(u_2 = 5 + 3 = 8\)
\(u_3 = 8 + 3 = 11\)
3. Sigma Notation (Section 4.3)
The symbol \(\Sigma\) (Sigma) is just a fancy Greek "S" that stands for Sum. It tells us to add up a bunch of terms.
\( \sum_{r=1}^{n} u_r \)
This tells you to:
- Start with \(r = 1\).
- Calculate every term up to \(n\).
- Add them all together.
Did you know? If you see \(\sum_{r=1}^{n} 1\), it literally means adding the number "1" to itself \(n\) times. So, the answer is just \(n\)!
4. Arithmetic Sequences and Series (Section 4.4)
An Arithmetic sequence is one where you add or subtract the same amount every time. This amount is called the common difference (\(d\)).
Key Formulas
\(n\)th term: \(u_n = a + (n-1)d\)
Sum of first \(n\) terms: \(S_n = \frac{n}{2}(2a + (n-1)d)\)
(Where \(a\) is the first term and \(d\) is the common difference).
The Proof
You need to know how to prove the sum formula! The trick is to write the sum out twice: once forwards and once backwards. When you add the two lines together, every pair of terms adds up to the same value \((2a + (n-1)d)\).
Key Takeaway:
If the gap between numbers is always the same, it's Arithmetic. Use \(d\) for the "Difference."
5. Geometric Sequences and Series (Section 4.5)
A Geometric sequence is one where you multiply by the same amount every time. This multiplier is called the common ratio (\(r\)).
Key Formulas
\(n\)th term: \(u_n = ar^{n-1}\)
Sum of first \(n\) terms: \(S_n = \frac{a(1 - r^n)}{1 - r}\)
Sum to Infinity (\(S_\infty\))
If the multiplier \(r\) is between -1 and 1 (written as \(|r| < 1\)), the terms get smaller and smaller until they basically disappear. We say the series converges.
\( S_\infty = \frac{a}{1 - r} \)
Analogy: If you eat half a pizza today, then half of what's left tomorrow, and keep going forever, you will eventually have eaten exactly 1 whole pizza. You never eat more than that! That's convergence.
Using Logs to find \(n\)
If you need to find out how many terms (\(n\)) it takes for a sum to exceed a certain value, you will often end up with an equation like \(r^n > k\). Use logarithms to solve for \(n\). Just remember that if you divide by the log of a number between 0 and 1, the inequality sign flips!
Key Takeaway:
Geometric sequences grow (or shrink) very fast. Look for a percentage increase or decrease in word problems—that's a sign it's geometric.
6. Modelling with Sequences (Section 4.6)
This is where you apply everything to real life!
- Arithmetic Modelling: Saving a fixed amount of money every month (e.g., £50, £100, £150...).
- Geometric Modelling: Compound interest in a bank, or a ball bouncing to a certain percentage of its previous height.
Common Mistake to Avoid:
Read the question carefully to see if the first term (\(a\)) happens at time \(t=0\) or \(t=1\). In many financial problems, the "first term" is the value after the first year, not the initial deposit.
Key Takeaway:
Always identify \(a\) and \(d\) (for arithmetic) or \(a\) and \(r\) (for geometric) first. Once you have those, you can solve almost any problem!