Sequences and series

Welcome to Sequences and Series!

In this chapter, we are going to explore the world of patterns. A sequence is simply a list of numbers following a specific rule, and a series is what happens when we add those numbers together. Whether it’s calculating how interest grows in a bank account or predicting the path of a bouncing ball, these mathematical tools are incredibly powerful. Don't worry if it seems like a lot of formulas at first—we’ll break them down into simple steps!

1. Understanding Sequences

A sequence is like a story where every number has its own place. We usually call the terms \(u_1, u_2, u_3 \dots\) where the small number at the bottom (the subscript) tells you the position.

Finding the nth Term

There are two main ways to describe a sequence:

• Deductive rules: A formula for the \(n\)th term, like \(u_n = 3n + 1\). If you want the 10th term, you just plug in \(n = 10\).
• Recurrence relations: A rule that tells you how to get to the next term from the current one. It looks like \(u_{n+1} = f(u_n)\). For example, \(u_{n+1} = u_n + 5\). This is like saying, "To get the next number, just add 5 to the one you have."

Types of Behavior

Sequences can behave in different ways:

• Increasing: Every term is bigger than the one before (\(u_{n+1} > u_n\)).
• Decreasing: Every term is smaller than the one before (\(u_{n+1} < u_n\)).
• Periodic: The terms repeat in a cycle (e.g., \(1, 2, 3, 1, 2, 3 \dots\)). The order is how many terms are in one cycle.

Quick Tip: If a question asks for a "recurrence relation," always look for the \(u_n\) and \(u_{n+1}\) symbols!

Key Takeaway: Sequences are just lists with rules. Deductive rules let you jump to any term; recurrence relations require you to step through one by one.

2. Sigma Notation \(\sum\)

The symbol \(\sum\) (the Greek letter Sigma) is just a fancy way of saying "Sum them all up!"

It looks like this: \(\sum_{r=1}^{n} f(r)\)

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

Example: \(\sum_{r=1}^{3} r^2\) means \(1^2 + 2^2 + 3^2 = 1 + 4 + 9 = 14\).

3. 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 first term is always called \(a\).

The Key Formulas

• The \(n\)th term: \(u_n = a + (n-1)d\)
• The sum of the first \(n\) terms (\(S_n\)): \(S_n = \frac{n}{2}[2a + (n-1)d]\)
• The "Short" Sum formula: \(S_n = \frac{n}{2}(a + L)\), where \(L\) is the very last term.

Common Mistake: When using the \(n\)th term formula, remember it’s \((n-1)\). If you want the 10th term, you only add the difference 9 times!

Key Takeaway: Use AP formulas for sequences that go up or down by a constant addition/subtraction, like a salary that increases by £500 every year.

4. Geometric Progressions (GP)

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

The Key Formulas

• The \(n\)th term: \(u_n = ar^{n-1}\)
• The sum of the first \(n\) terms: \(S_n = \frac{a(1-r^n)}{1-r}\) (or \(\frac{a(r^n-1)}{r-1}\))

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

Some GPs get smaller and smaller until the numbers being added are so tiny they almost don't matter. This happens only when the modulus of \(r\) is less than 1 (\(|r| < 1\)), meaning \(r\) is between -1 and 1.

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

Did you know? This is why a bouncing ball eventually stops. Each bounce is a fraction of the height of the previous one (a GP), and the total distance traveled is a "sum to infinity" problem!

Key Takeaway: GPs involve multiplication. If the terms are getting smaller (\(|r| < 1\)), you can calculate a total "limit" called the sum to infinity.

5. Binomial Expansion

The Binomial Expansion is a way of multiplying out brackets like \((a + bx)^n\) without having to do it manually for hours!

Case A: Positive Integer \(n\)

When \(n\) is a whole number (like 2, 3, 4...), we use Pascal's Triangle or the nCr button on your calculator.

The notation \(\binom{n}{r}\) or \(^nC_r\) tells you how many ways to choose \(r\) items from \(n\). This is linked to binomial probabilities in statistics!

Case B: Any Rational \(n\) (Negative or Fractions)

If \(n\) is a fraction or a negative number, the expansion never ends! We use this formula for \((1+x)^n\):

\((1+x)^n = 1 + nx + \frac{n(n-1)}{2!}x^2 + \frac{n(n-1)(n-2)}{3!}x^3 + \dots\)

CRITICAL RULE: This expansion is only valid (meaning it actually works) if \(|x| < 1\). If your bracket is \((a + bx)^n\), the rule is \(|\frac{bx}{a}| < 1\).

Memory Aid: Factorials (\(n!\)) are just "excited" numbers! \(4!\) is \(4 \times 3 \times 2 \times 1 = 24\). Always use them in the denominators of binomial expansions.

Step-by-Step for \((a+bx)^n\):
1. Factor out the \(a\) so the bracket starts with a 1: \(a^n(1 + \frac{bx}{a})^n\).
2. Apply the formula to the bracket.
3. Multiply everything by the \(a^n\) at the end.

Key Takeaway: Use binomial expansion to turn complex brackets into a series of simple terms. Always check the validity range for negative or fractional powers!

6. Modelling with Sequences

In the exam, you might be asked to apply these to "real life."

• Arithmetic: Simple interest, saving a fixed amount each month, stadium seating rows.
• Geometric: Compound interest, population growth (where it grows by a percentage), radioactive decay.

Quick Review Box:
- AP: Add \(d\) each time. Look for "constant increase."
- GP: Multiply by \(r\). Look for "% increase" or "ratio."
- \(\sum\): Add them all up.
- Validity: Only for Binomials where the power isn't a positive whole number.

Don't worry if this seems tricky at first! The best way to master sequences and series is to practice identifying whether a pattern is Arithmetic (adding) or Geometric (multiplying). Once you know that, you just pick the right formula and plug in the numbers!