Sequences are 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 use previous terms to find the next one.

Factorials, written as n!, are products of whole numbers from 1 to n. They show up in many sequence formulas and can even be the terms of a sequence themselves. Understanding these concepts helps in analyzing sequence behavior and summation.

Sequences and Their Notations

Explicit formulas for sequences

• Allows direct calculation of any term's value in a sequence
  • Uses the notation $a_n$ where $n$ represents the term's position (1st, 2nd, 3rd, etc.)
  • For the sequence 2, 5, 8, 11, ..., the explicit formula is $a_n = 3n - 1$ (plugging in $n=1$ gives the 1st term 2, $n=2$ gives the 2nd term 5, and so on)
• Arithmetic sequences have a constant difference between consecutive terms ($+3$ for 2, 5, 8, 11, ...)
  • Explicit formula: $a_n = a_1 + (n - 1)d$ where $a_1v$ is the first term and $d$ is the common difference
  • For 2, 5, 8, 11, ..., $a_1=2$ and $d=3$, so the formula is $a_n = 2 + (n-1)3 = 3n - 1$
• Geometric sequences have a constant ratio between consecutive terms ($\times 2$ for 1, 2, 4, 8, 16, ...)
  • 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, ..., $a_1=1$ and $r=2$, so the formula is $a_n = 1 \cdot 2^{n-1} = 2^{n-1}$

Recursive formulas for sequences

• Defines each term using one or more previous terms
  • Uses the notation $a_n = f(a_{n-1}, a_{n-2}, ...)$ where $f$ is a function of the previous terms
  • The Fibonacci sequence 0, 1, 1, 2, 3, 5, ... has the recursive formula $a_n = a_{n-1} + a_{n-2}$ for $n \geq 3$ with $a_1 = 0$ and $a_2 = 1$
• Generating terms with a recursive formula:
  1. Start with the given initial terms ($a_1 = 0$ and $a_2 = 1$ for Fibonacci)
  2. Apply the recursive formula to find each subsequent term ($a_3 = a_2 + a_1 = 1+0=1$, $a_4 = a_3 + a_2 = 1+1=2$, $a_5 = a_4 + a_3 = 2+1=3$, and so on for Fibonacci)

Factorial notation in sequences

• Factorial notation $n!$ is the product of all positive integers from 1 to $n$
  • $5! = 5 \times 4 \times 3 \times 2 \times 1 = 120$
  • $0! = 1$ by definition
• Factorials appear in sequence formulas like:
  • Number of permutations of $n$ distinct objects: $P(n) = n!$ (arranging 5 books on a shelf can be done in $5! = 120$ ways)
  • Number of combinations of $n$ objects taken $r$ at a time: $C(n,r) = \frac{n!}{r!(n-r)!}$ (choosing 3 toppings from 5 options can be done in $\frac{5!}{3!(5-3)!} = \frac{5!}{3!2!} = 10$ ways)
• Factorials can also be terms in a sequence
  • The sequence 1, 2, 6, 24, 120, ... has the explicit formula $a_n = n!$ ($a_1 = 1! = 1$, $a_2 = 2! = 2$, $a_3 = 3! = 6$, $a_4 = 4! = 24$, $a_5 = 5! = 120$)

Sequence Behavior and Summation

• Sequence convergence occurs when terms approach a specific value as n increases
• Sequence divergence happens when terms do not approach a specific value or grow without bound
• Limit of a sequence represents the value that terms approach as n approaches infinity
• Infinite series are formed by adding all terms of an infinite sequence
• Summation notation (∑) is used to represent the sum of a finite or infinite series of terms