13.1 Sequences and Their Notations

Sequences and Their Notations

Sequences are ordered lists of numbers that follow specific patterns. They show up throughout math as tools for modeling real-world scenarios, from population growth to loan payments. This topic covers how to generate and work with sequences using explicit formulas, recursive definitions, and factorial notation.

Generation of Explicit Sequence Terms

An explicit formula gives you a direct way to calculate any term in a sequence without knowing the terms before it. You write it as $a_n = f(n)$, where $n$ is the term's position number.

For example, the sequence 2, 5, 8, 11, ... has the explicit formula $a_n = 3n - 1$. To find any term, just plug in the position number for $n$:

• 1st term: $a_1 = 3(1) - 1 = 2$
• 10th term: $a_{10} = 3(10) - 1 = 29$
• 100th term: $a_{100} = 3(100) - 1 = 299$

That's the power of explicit formulas: you can jump straight to the 100th term without calculating the first 99.

Two Common Types

Arithmetic sequences have a constant difference between consecutive terms. Their explicit formula is:

$a_n = a_1 + (n - 1)d$

where $a_1$ is the first term and $d$ is the common difference. In the sequence 2, 5, 8, 11, ..., the common difference is 3 because each term is 3 more than the previous one.

Geometric sequences have a constant ratio between consecutive terms. Their explicit formula is:

$a_n = a_1 \cdot r^{n-1}$

where $a_1$ is the first term and $r$ is the common ratio. In the sequence 2, 6, 18, 54, ..., the common ratio is 3 because each term is 3 times the previous one.

A quick way to tell them apart: if you're adding the same number each time, it's arithmetic. If you're multiplying by the same number each time, it's geometric.

Construction of Recursive Sequence Terms

A recursive formula defines each term using one or more previous terms. Unlike explicit formulas, you can't jump directly to a term; you have to build up to it step by step.

Every recursive formula needs two things:

1. Initial condition(s): the starting term(s) that anchor the sequence
2. Recursion rule: the formula that relates each new term to previous terms

Without initial conditions, the sequence is undefined because there's nothing to start from.

Example: The Fibonacci Sequence

The Fibonacci sequence (0, 1, 1, 2, 3, 5, 8, ...) is defined recursively:

• Initial conditions: $a_1 = 0$, $a_2 = 1$
• Recursion rule: $a_n = a_{n-1} + a_{n-2}$ for $n \geq 3$

To find the 6th term, you calculate each term in order:

1. $a_3 = a_2 + a_1 = 1 + 0 = 1$
2. $a_4 = a_3 + a_2 = 1 + 1 = 2$
3. $a_5 = a_4 + a_3 = 2 + 1 = 3$
4. $a_6 = a_5 + a_4 = 3 + 2 = 5$

Notice that you need every prior term to get to the one you want. This is the main tradeoff: recursive formulas are often easier to write, but harder to compute for large $n$.

Arithmetic and geometric sequences can also be written recursively:

• Arithmetic: $a_n = a_{n-1} + d$, with $a_1$ given
• Geometric: $a_n = r \cdot a_{n-1}$, with $a_1$ given

Application of Factorial Notation

Factorial notation (written with !) means you multiply a positive integer by every positive integer below it:

$5! = 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1 = 120$

A few more examples to build intuition:

• $3! = 3 \cdot 2 \cdot 1 = 6$
• $4! = 4 \cdot 3 \cdot 2 \cdot 1 = 24$
• $1! = 1$

By definition, $0! = 1$. This might seem odd, but it keeps formulas in combinatorics and probability consistent.

Why Factorials Matter

The number of ways to arrange $n$ distinct objects in order (called permutations) is $n!$. For instance, 3 books can be arranged on a shelf in $3! = 6$ different orders.

Simplifying Factorial Expressions

You'll often need to simplify fractions involving factorials. The key trick is to expand and cancel:

$\frac{n!}{(n-k)!} = n \cdot (n-1) \cdot (n-2) \cdots (n-k+1)$

For example: $\frac{7!}{4!} = \frac{7 \cdot 6 \cdot 5 \cdot 4!}{4!} = 7 \cdot 6 \cdot 5 = 210$

The binomial coefficient combines factorials in a specific way:

$\binom{n}{k} = \frac{n!}{k!(n-k)!}$

This counts the number of ways to choose $k$ objects from a set of $n$ objects when order doesn't matter. For example, choosing 2 people from a group of 5: $\binom{5}{2} = \frac{5!}{2! \cdot 3!} = \frac{120}{2 \cdot 6} = 10$.

Sequences Involving Factorials

Factorials can appear in sequence formulas. The simplest example is the sequence of factorials itself:

$1!, \; 2!, \; 3!, \; 4!, \; 5!, \; \ldots = 1, \; 2, \; 6, \; 24, \; 120, \; \ldots$

These values grow extremely fast, which is why factorial-based sequences come up in probability and counting problems.

Behavior of Sequences and Series

Once you can generate terms of a sequence, the next question is: what happens as the sequence goes on?

Convergence means the terms of a sequence approach a specific value (called the limit) as $n$ gets larger and larger. For example, the sequence $a_n = \frac{1}{n}$ produces 1, 0.5, 0.333..., 0.25, ... and converges to 0.

Divergence means the terms don't settle toward any single value. The sequence $a_n = 2n$ produces 2, 4, 6, 8, ... and diverges because the terms keep growing without bound.

A series is what you get when you add up the terms of a sequence. Series are often written in summation notation (using the $\Sigma$ symbol), which compactly expresses the sum. For example:

$\sum_{n=1}^{4} n^2 = 1^2 + 2^2 + 3^2 + 4^2 = 30$

The distinction matters: a sequence is a list of numbers, while a series is the sum of those numbers.