11.1 Sequences and Their Notations

Sequences and Their Notations

Sequences are ordered lists of numbers that follow specific patterns. They can be described using explicit formulas, which let you find any term directly, or recursive formulas, which build each term from previous ones. Factorials ($n!$) show up frequently in sequence formulas and throughout counting theory. Understanding how to write, interpret, and evaluate these formulas is the foundation for everything else in this unit.

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Explicit Formulas for Sequences

An explicit formula lets you jump straight to any term in a sequence without calculating all the ones before it. You plug in the term number $n$, and out comes the value.

• The notation $a_n$ represents the term at position $n$ (so $a_1$ is the first term, $a_2$ is the second, etc.)
• For the sequence 2, 5, 8, 11, ..., the explicit formula is $a_n = 3n - 1$. Plugging in $n = 1$ gives 2, $n = 2$ gives 5, and so on.

Arithmetic sequences have a constant difference between consecutive terms.

• Explicit formula: $a_n = a_1 + (n - 1)d$, where $a_1$ is the first term and $d$ is the common difference
• For 2, 5, 8, 11, ..., the common difference is $d = 3$ and $a_1 = 2$, so $a_n = 2 + (n-1)(3) = 3n - 1$

Geometric sequences have a constant ratio between consecutive terms.

• Explicit formula: $a_n = a_1 \cdot r^{n-1}$, where $a_1$ is the first term and $r$ is the common ratio
• For 1, 2, 4, 8, 16, ..., the common ratio is $r = 2$ and $a_1 = 1$, so $a_n = 1 \cdot 2^{n-1} = 2^{n-1}$

The key distinction: arithmetic means you're adding the same amount each time, geometric means you're multiplying by the same amount each time.

Recursive Formulas for Sequences

A recursive formula defines each term using one or more previous terms. You can't jump to the 50th term directly; you have to work your way there from the initial term(s).

• The general notation is $a_n = f(a_{n-1}, a_{n-2}, ...)$, where $f$ is some function of earlier terms
• Every recursive formula needs initial conditions (starting values), or you'd have nothing to build from

The Fibonacci sequence is the classic example: 0, 1, 1, 2, 3, 5, 8, ... It's defined by $a_n = a_{n-1} + a_{n-2}$ for $n \geq 3$, with $a_1 = 0$ and $a_2 = 1$.

Generating terms from a recursive formula:

1. Write down the given initial terms (for Fibonacci: $a_1 = 0$, $a_2 = 1$)
2. Apply the formula to find the next term: $a_3 = a_2 + a_1 = 1 + 0 = 1$
3. Keep applying: $a_4 = a_3 + a_2 = 1 + 1 = 2$, then $a_5 = a_4 + a_3 = 2 + 1 = 3$, and so on

Arithmetic and geometric sequences can also be written recursively. For example, the arithmetic sequence 2, 5, 8, 11, ... can be written as $a_n = a_{n-1} + 3$ with $a_1 = 2$. The geometric sequence 1, 2, 4, 8, ... can be written as $a_n = 2a_{n-1}$ with $a_1 = 1$.

Factorial Notation in Sequences

Factorial notation $n!$ means the product of all positive integers from 1 to $n$.

• $5! = 5 \times 4 \times 3 \times 2 \times 1 = 120$
• $3! = 3 \times 2 \times 1 = 6$
• $1! = 1$
• $0! = 1$ by definition (this one you just have to memorize)

Factorials grow extremely fast, which is why they're so useful in counting problems.

Factorials appear in key counting formulas:

• Permutations of $n$ distinct objects: $P(n) = n!$ (arranging 5 books on a shelf can be done in $5! = 120$ ways, since order matters)
• Combinations of $n$ objects taken $r$ at a time: $C(n, r) = \frac{n!}{r!(n-r)!}$ (choosing 3 toppings from 5 options gives $\frac{5!}{3! \cdot 2!} = 10$ ways, since order doesn't matter)

Factorials can also define the terms of a sequence directly. The sequence 1, 2, 6, 24, 120, ... has the explicit formula $a_n = n!$.

Sequence Behavior and Summation

As you look at more and more terms of a sequence, two things can happen:

• Convergence: The terms approach a specific value as $n$ increases. For example, $a_n = \frac{1}{n}$ gives 1, 0.5, 0.333..., 0.25, ... which approaches 0. The limit of this sequence is 0.
• Divergence: The terms don't settle toward any single value. They might grow without bound (like $a_n = 2^n$) or oscillate (like $a_n = (-1)^n$).

Summation notation uses the symbol $\sum$ to compactly represent the sum of terms in a sequence. For example:

$\sum_{n=1}^{4} (3n - 1) = 2 + 5 + 8 + 11 = 26$

This tells you to evaluate $3n - 1$ for $n = 1, 2, 3, 4$ and add the results. When you extend this to infinitely many terms, you get an infinite series, which only has a finite sum if the underlying sequence converges in the right way.