Introduction to Sequences

What are sequences?

• A sequence represents an ordered collection of numbers.
• Each individual number within a sequence is termed a "term".
• The position of a term within a sequence is identified as its "position". The symbol 'n' is commonly utilized to denote an unknown position.
• Subscript notation is employed to refer to a specific term. For instance, 'a₁' denotes the first term, 'a₇' signifies the seventh term, and 'a_n' represents the n^th term.

What is a position-to-term rule?

• A position-to-term rule provides the n^th term of a sequence based on the value of n. 
• This mathematical concept is highly advantageous as it allows us to determine the 100^th term of a sequence without calculating the preceding 99 terms.

What is a term-to-term rule?

• A term-to-term rule gives the (n + 1)^th term in terms of the n^th term
  • ie a_n+1 is given in terms of an
  • If a term is known, the next one can be worked out

How do I use the position-to-term and term-to-term rules?

• Position-to-term and term-to-term rules are essential for generating sequences.
• By analyzing a given sequence, we can deduce these rules and predict subsequent terms.
• Understanding different types of sequences is crucial, such as:
  • Linear and quadratic sequences
  • Geometric sequences
  • Fibonacci sequences
  • Other sequences