Section 11.1

Sequences & Notation

This section gives you the vocabulary and notation for the rest of the chapter. Arithmetic series, geometric series, and the Binomial Theorem all use the $a_n$ notation and ideas you learn here — so getting comfortable now saves you a lot of confusion later.

⚡ Quick Summary

A sequence is just a list of numbers that follow a pattern — like 2, 4, 6, 8, ... or 1, 3, 9, 27, ... Think of it like a playlist: each song has a position (track 1, track 2, etc.), and each number in the sequence has a position too. We write $a_1$ for the first number, $a_2$ for the second, and $a_n$ for whichever position we care about.

Explicit Formula

a_n = f(n)

Gives the nth term directly as a function of n — no previous terms needed

Recursive Formula

a_n = f(a_{n-1}), \quad a_1 = \text{(given)}

Each term depends on the previous term; you must know the starting value

📋 Before You Start

Tap any item if you need a refresher:

Plain-English version

There are two ways to describe a pattern. An explicit formula is like GPS — it takes you directly to any term you want. Give it position 100 and it hands you the answer immediately. A recursive formula is more like walking directions — "start here, then take one step forward each time." It tells you how to get the *next* term from the *previous* one, so you have to build up step by step.

Factorials ( $n!$ ) are a shorthand for "multiply all the whole numbers from 1 up to $n$ ." So $4! = 4 \times 3 \times 2 \times 1 = 24$ . They grow shockingly fast — $10!$ is already 3.6 million! One quirky rule: $0! = 1$ . It seems weird, but it's a convention that keeps all the formulas working neatly. You'll see factorials again when you get to the Binomial Theorem.

When you're given a list of numbers and asked to find the pattern, check two things: are the differences between terms constant (like +3 every time)? Or are the ratios constant (like ×2 every time)? The first is called arithmetic, the second geometric — and you'll study both in the next sections.

Key Formulas

Explicit Formula

a_n = f(n)

Gives the nth term directly as a function of n — no previous terms needed

Recursive Formula

a_n = f(a_{n-1}), \quad a_1 = \text{(given)}

Each term depends on the previous term; you must know the starting value

Factorial

n! = n \times (n-1) \times (n-2) \times \cdots \times 2 \times 1, \qquad 0! = 1

Grows extremely fast; 0! = 1 by definition

Key Takeaways

✓Explicit formulas ( $a_n = f(n)$ ) let you jump to any term; recursive formulas require building from the start
✓Factorials grow faster than any exponential — $n!$ wins over $a^n$ eventually
✓ $0! = 1$ by definition — this makes combination formulas work correctly
✓To find a formula from a list, check for a constant difference (arithmetic) or constant ratio (geometric)

⚠️ Common Mistakes

✗Confusing explicit and recursive formulas — if the formula references $a_{n-1}$ , it's recursive and you need the previous term; if it only has $n$ , it's explicit
✗Starting with the wrong index — $a_1$ means plug in $n = 1$ , not $n = 0$ , unless the problem specifically says otherwise
✗Forgetting that $0! = 1$ by definition — it looks weird but it's a convention that makes all the combination and binomial formulas work
✗Computing $n!$ fully instead of canceling — $\frac{8!}{5!}$ doesn't require computing $8! = 40320$ ; just cancel the $5!$ to get $8 \times 7 \times 6 = 336$

Next: Section 11.2

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10.2 The Hyperbola

11.2 Arithmetic Sequences