A sequence is just a list of numbers,  a₀, a₁, a₂, ...

We sometimes use the notation  {a_n}  for the entire sequence. We call  a_n  the general term of the sequence. The subscript  n  is the index, and tells us which term to consider.

We can think of any sequence as a function  f  from the indicies (the natural numbers  0, 1, 2, ...) to the numbers in the sequence. Thus, f(n) = a_n . The input is the place in the sequence; the output is the term that we will find there. We often look for patterns in sequences of numbers, and then try to express that pattern as a formula for the general term.

Arithmetic sequences have an especially simple pattern: Each term comes from adding the same constant to the previous term. Thus, if  a₀  is the first (zero^th) term of the sequence, subsequent terms will be  a₀ + r , a₀ + 2r , a₀ + 3r , etc., for some constant  r . The pattern is  f(n) = a₀ + nr . The function is linear.

Arithmetic sequences are found in situations where there is constant accumulation. Examples include drops from a faucet into a sink, dust settling, and regular deposits of equal size into a bank account with no interest.

Geometric sequences are similar to arithmetic ones, but each term is a constant multiple, rather than a constant interval, from the previous term. Thus, if  a₀  is the first (zero^th) term of the sequence, subsequent terms will be  a₀ r , a₀ r ² , a₀ r ³ , etc., for some constant  r . The pattern is  f(n) = a₀ r ⁿ . The function is exponential.

Geometric sequences are found in situations where there is proportional accumulation. Examples include the thickness of a piece of paper that you keep folding in half, populations that double in size after a certain amount of time, and a bank deposit accumulating interest at a fixed rate.