Sequences represent ordered lists of elements and can be defined as a function from a subset of natural numbers to a set. There are two main types of sequences: arithmetic sequences, where each term is obtained by adding a common difference to the previous term, and geometric sequences, where each term is obtained by multiplying the previous term by a common ratio. Mathematical induction can be used to prove properties are true for all terms in a sequence by showing the base case holds and the inductive step follows from the assumption.