Finite Sequences

A sequence is a list of elements with a particular order. While the idea of a sequence of numbers, a , a , · · · , is straight forward, it is useful to think of a sequence as a function whose domain is either the set of first n natural numbers or N . Throughout this chapter, we consider only sequences of real numbers and we will refer to them as sequences. The arithmetic sequences and geometric sequences are also known as arithmetic progressions(AP) and geometric progressions (GP).

Let us recall, the basic definitions of sequences and series. • If X is any set and n ∈ N , then any function f : { , , , . . .

, n } → X is called a finite sequence on X and any function g : N → X is called an infinite sequence on X . The value f ( n ) of the function f at n is denoted by a n and the sequence itself is denoted by ( a n ) . • If the set X happens to be a set of real numbers, the sequence is called a numerical sequence or a sequence of real numbers. • Though every sequence is a function, a function is not necessarily a sequence.

• Unlike sets, where elements are not repeated, the terms in a sequence may be repeated. In particular, a sequence in which all terms are same is called a constant sequence. • A useful way to visualise a sequence ( a n ) is to plot the graph of { ( n, a n ) : n ∈ N } which gives some details about the sequence. .

Finite Sequences Figure . . .