Sequences

Let us start off this section along with a discussion of just what a sequence is. A sequence is nothing much more than a list of numbers written in a particular order. The list may or may not consist of an infinite number of terms in them even though we will be dealing exclusively with infinite sequences in this class.  Common sequence terms are represented as follows,

a1 - first term

a2 - second term .....

a_n  - n^th  term

a_n+1- (n+1)^st term

As we will be dealing with infinite sequences every term in the sequence will be followed by other term as described above.  In the notation above we require to be very cautious with the subscripts. The subscript of n + 1 represents the next term in the sequence and NOT the one plus the n^th term!  Alternatively,

A_n+1 ≠ a_n+1

Thus should be very careful while writing subscripts to ensure that the "+1" doesn't migrate out of the subscript! This is an simple mistake to make while you first start dealing with this type of thing.

There is a range of ways of that representing a sequence. Each of the following is similar ways of representing a sequence.

{a₁, a₂, ......, a_n, a_n+1, ...}            

{a_n}             

{an}^∞ _n=1

In the above second and third notations is generally given by a formula.

A pair of notes is now in order about these notations.  First, note the variation among the above second and third notations.  If the starting point is not significant or is implied in some way through the problem it is frequently not written down as we did in the third notation.  Subsequently, we utilized a starting point of n = 1 in the third notation only thus we could write one down. Totally there is no reason to believe that a sequence will start at n = 1 .  A sequence will begin where ever it require to start.