13.1 Sequences and Their Notations

4 min read•june 24, 2024

Sequences are ordered lists of numbers following specific patterns. They're crucial in math for modeling real-world scenarios and solving complex problems. This topic covers how to generate and work with sequences using explicit formulas, recursive definitions, and .

Understanding sequences helps us analyze patterns and make predictions. We'll explore different types of sequences, like arithmetic and geometric, and learn how to find specific terms. We'll also dive into factorials and their applications in combinatorics and probability.

Sequences and Their Notations

Generation of explicit sequence terms

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An for a is a function that allows direct calculation of the , usually written as
- For the sequence 2, 5, 8, 11, ... the is $a_n = 3n - 1$
To find the nth term using the explicit formula, substitute the desired term number for n in the function
- To find the 10th term of the sequence 2, 5, 8, 11, ... use $a_{10} = 3(10) - 1 = 29$
Common types of sequences with explicit formulas:
- Arithmetic sequences: , where $a_1$ $a_{1}$ is the first term and d is the
  - In an , the difference between consecutive terms is constant (e.g., 2, 5, 8, 11, ... has a common difference of 3)
- Geometric sequences: , where $a_1$ $a_{1}$ is the first term and r is the
  - In a , each term is a constant multiple of the previous term (e.g., 2, 6, 18, 54, ... has a of 3)

Construction of recursive sequence terms

A defines each term of a sequence using one or more of the previous terms
- For the 0, 1, 1, 2, 3, 5, ... the recursive formula is $a_n = a_{n-1} + a_{n-2}$ for $n \geq 3$ , with $a_1 = 0$ and $a_2 = 1$
To find the nth term using a recursive formula, calculate each term sequentially until reaching the desired term
- To find the 6th term of the Fibonacci sequence, calculate:
  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. $a_6 = a_5 + a_4 = 3 + 2 = 5$
Recursive formulas require , which specify the starting term(s) of the sequence
- Initial conditions are necessary because the recursive formula depends on previous terms
- Without initial conditions, the sequence would be undefined

Application of factorial notation

Factorial notation, denoted by !, represents the product of a positive integer and all positive integers less than it
- $5! = 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1 = 120$
By definition, $0! = 1$ $0! = 1$
- This is a special case that allows for consistency in mathematical formulas and combinatorial identities
Factorials are often used in sequence formulas, particularly in combinatorics and probability
- The number of of n distinct objects is $n!$ $n!$
  - A permutation is an arrangement of objects in a specific order
Simplifying expressions with factorials:
- $\frac{n!}{(n-k)!} = n \cdot (n-1) \cdot (n-2) \cdot ... \cdot (n-k+1)$ $\frac{n !}{( n - k )!} = n \cdot (n - 1) \cdot (n - 2) \cdot ... \cdot (n - k + 1)$ , where $n \geq k$ $n \geq k$
  - This simplification is useful when working with permutations and combinations
- $\frac{n!}{k!(n-k)!} = \binom{n}{k}$ $\frac{n !}{k ! ( n - k )!} = (k n)$ , the
  - The binomial coefficient represents the number of ways to choose k objects from a set of n objects
Sequences involving factorials:
- The sequence of factorials: 1, 1, 2, 6, 24, 120, ...
  - Each term is the factorial of its position (1!, 2!, 3!, 4!, 5!, ...)
- The sequence of (permutations with no fixed points): 1, 0, 1, 2, 9, 44, 265, ...
  - A derangement is a permutation where no element appears in its original position
  - The nth term of this sequence is given by

Behavior of Sequences and Series

A sequence is an ordered list of numbers that follow a specific pattern or rule
occurs when the terms of a sequence approach a specific value () as n approaches infinity
happens when a sequence does not converge to a specific value
The limit of a sequence, if it exists, is the value that the terms approach as n increases indefinitely
A is the sum of the terms of a sequence
- Series can be represented using summation notation, which compactly expresses the sum of sequence terms