Sequences and series

Welcome to Sequences and Series!

Ever noticed the pattern in a pile of oranges at the supermarket or the way a bouncing ball slowly comes to a stop? In this chapter, we explore sequences (ordered lists of numbers) and series (the sum of those numbers). These tools help us predict future values and calculate totals in everything from finance to physics. 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 simply a list of numbers following a specific rule. Each number in the list is called a term. We usually use \(n\) to represent the position of the term (1st, 2nd, 3rd...).

Formula for the \(n\)th Term

Sometimes, we have a formula where we can just plug in the position \(n\) to find the value.
Example: If the rule is \(x_n = 2n + 3\), the 1st term (\(n=1\)) is \(2(1) + 3 = 5\).

Recurrence Relations

Other times, a term depends on the one before it. This is written as \(x_{n+1} = f(x_n)\). Think of this as a "Step-by-Step" rule: "To get the next number, do something to the current one."

Finding the Limit (\(L\))

Some sequences eventually settle down to a single number as \(n\) gets very large. This is called the limit.
The Trick: To find the limit \(L\), we assume that eventually, the terms stop changing. So, we replace both \(x_{n+1}\) and \(x_n\) with \(L\) in the formula and solve for \(L\).
\(L = f(L)\)

Quick Review:
• \(x_n\) is the term at position \(n\).
• \(x_{n+1}\) is the "next" term.
• To find a limit, set \(next = current = L\).

2. Arithmetic Series

An Arithmetic sequence is one where you add (or subtract) the same amount each time. This amount is called the common difference (\(d\)).

Key Terms:

• \(a\): The first term.
• \(d\): The common difference.
• \(n\): The number of terms.

The Formulas:

1. The \(n\)th term: \(u_n = a + (n - 1)d\)
2. The sum of the first \(n\) terms (\(S_n\)): \(S_n = \frac{n}{2}[2a + (n - 1)d]\)

Analogy: Imagine walking up a staircase where every step is exactly 15cm high. Your height increases "arithmetically" by 15cm with every step.

Sigma (\(\sum\)) Notation

The giant Greek letter \(\sum\) just means "add them all up."
Example: \(\sum_{r=1}^{4} 2r\) means "find the value of \(2r\) for \(r=1, 2, 3, 4\) and add them together": \(2(1) + 2(2) + 2(3) + 2(4) = 20\).

Common Mistake: When finding \(d\), always do (Term 2 minus Term 1), not the other way around. If the numbers are going down, \(d\) must be negative!

3. Geometric Series

A Geometric sequence is one where you multiply by the same amount each time. This is called the common ratio (\(r\)).

Did you know? Geometric sequences grow much faster than arithmetic ones. This is how "viral" videos spread—one person shares it with three, then those three share it with three more, and so on!

The Formulas:

1. The \(n\)th term: \(u_n = ar^{n-1}\)
2. The sum of a finite series: \(S_n = \frac{a(1 - r^n)}{1 - r}\)

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

If the multiplier \(r\) is a fraction between -1 and 1 (written as \(|r| < 1\)), the terms get smaller and smaller, eventually reaching zero. In this special case, you can add up infinitely many terms and get a finite total!
Formula: \(S_\infty = \frac{a}{1 - r}\)

Encouraging Phrase: Don't worry if "adding up forever" sounds impossible! Think of it like this: if you eat half a cake, then half of what's left, then half of that... you will eventually eat exactly 1 whole cake, but never more.

Key Takeaway:
• Arithmetic: Additive (\(d\)).
• Geometric: Multiplicative (\(r\)).
• Sum to Infinity: Only works if \(-1 < r < 1\).

4. The Binomial Expansion

The word "Binomial" means "two terms." This section is about how to expand brackets like \((1 + x)^n\) without multiplying them manually for hours.

Using Pascal’s Triangle

For small positive integers of \(n\), you can find the coefficients (the numbers in front of the terms) using the triangle:
Row 0: 1
Row 1: 1, 1
Row 2: 1, 2, 1
Row 3: 1, 3, 3, 1
Each number is the sum of the two numbers directly above it!

The Formula Method

For \((1 + x)^n\), the expansion is:
\(1 + nx + \frac{n(n-1)}{2!}x^2 + \dots\)

Important Terms:

• \(n!\) (Factorial): Multiply the number by every whole number below it. \(4! = 4 \times 3 \times 2 \times 1 = 24\).
• \(\binom{n}{r}\): This is often called "\(n\) choose \(r\)" and can be found on your calculator. It tells you the coefficient for a specific term.

Step-by-Step Expansion for \((a + b)^n\):
1. The powers of \(a\) start at \(n\) and go down to 0.
2. The powers of \(b\) start at 0 and go up to \(n\).
3. Use \(\binom{n}{r}\) or Pascal's Triangle to find the numbers to put in front.

Quick Review:
• \(n\) must be a positive integer in this section.
• The total number of terms in the expansion is always \(n + 1\).
• If you see \((1 - x)^n\), treat the second term as \((-x)\) and be careful with your signs!

Summary: Your "Cheat Sheet" for Success

Sequence Type: Arithmetic | Rule: \(+d\) | Key Formula: \(S_n = \frac{n}{2}[2a + (n-1)d]\)
Sequence Type: Geometric | Rule: \(\times r\) | Key Formula: \(S_n = \frac{a(1-r^n)}{1-r}\)
Convergent Geometric: If \(|r| < 1\), then \(S_\infty = \frac{a}{1-r}\)
Binomial: Use \(\binom{n}{r}\) or Pascal's Triangle for coefficients.

Keep practicing these formulas, and soon you'll be spotting patterns everywhere! You've got this!