• A sequence \(a_n\) is said to be bounded from above if there exists \(M\in\mathbb{R}\), called the upper bound, such that \(a_n\leq M\) for all \(n\).
• A sequence \(a_n\) is said to be bounded from above if there exists \(m\in\mathbb{R}\), called the lower bound, such that \(m\leq a_n\) for all \(n\).
• A sequence \(a_n\) is said to be bounded provided that it is both bounded from above and bounded from below; otherwise, the sequence is said to be unbounded

• A sequence \(a_n\) is said to be monotonic if it is either nondecreasing or nonincreasing.
• A sequence \(a_n\) is said to be nondecreasing provided that \(a_n\leq a_{n+1}\) for integers n. If the inequality is strict i.e. \(a_{n}<a_{n+1}\) then the sequence is said to be increasing, or strictly increasing.
• A sequence \(a_n\) is said to be nonincreasing provided that \(a_n\geq a_{n+1}\) for integers n. If the inequality is strict i.e. \(a_{n+1}<a_{n}\) then the sequence is said to be decreasing, or strictly decreasing
• A sequence \(a_n\) is said to be eventually monotonic if there exists an integer \(N\) such that the sequence is monotonic \(N\leq n\)
  

• A function \(f(x)\) is said to be a continuous extension for a sequence \(a_n\) provided that \(f(x)\) is a continuous function and the two functions are the same for each integer on the domain for \(a_n\). In other words, if  \(a_n = f (n)\) for each \(n\) then \(f(x),\) is a continuous extension for \(a_n\). 
• Caution: A continuous extension is not found by simplify substituting the discrete variable \(n\) for the continuous variable \(\)x./latex]. To illustrate this truth, consider the following examples. 

• Inspect the ratio and difference among consecutive terms.
• How do the terms toggle if they do toggle?
• What changes and what remains the same?
• Do not prematurely collapse the pattern.
  • Consider factoring numbers
  • Keep track of changes in exponents 
• Avoid summing and preserve iterations

Some Common Types of Sequences

1. Arithmetic Sequence
   • An arithmetic sequence is a sequence whose terms differ by a linear scaling, i.e. \(a_{n+1}=a_n+nk\) where \(k\) is some constant (it maybe an integer or non-integer).
2. Geometric Sequence
   • An arithmetic sequence is a sequence whose terms difference by exponentially. This types of sequences have the form \(a_{n+1}=ka_n\) where \(k\) is some constant (it maybe an integer or non-integer).
3. Recursive Sequence
   • A recursive sequence (also known as a recursion) is a sequence whose terms are defined by its previous terms. These sequences can be linear such as \(a_{n+2}=a_{n+1}+2a_{n}\) or they non-linear such as \(a_{n+2}=a_na_{n+1}\). these types of sequences of studied in great detail in later courses such as Discrete Mathematics.