Sequences

Recall that when we defined sets, we stated that sets like \(\{0,1,2,3,4,5,…\}\) and \(\{0,2,1,3,4,6,5,7,…\}\) were equal because they contained the same elements. However, as sequences these are not equal because, while they have the same elements, they are in a different order. With sequences, order matters; that is, we care about which element is listed first, second, third, etc.

More often than not, we will be dealing with infinite sequences. In fact, some authors define sequences to be an infinite enumerable linearly ordered set.

Representing Sequences

As seen above, we can represent sequences as a list of elements between either curly brackets or round braces. For instance, the sequences below are equal, but differ only in notation.

$$\{2,56,3,6,0,-12,…\}$$

$$(2,56,3,6,0,-12,…)$$

When dealing with abstract symbolically represented sequences where we don’t necessarily know what the elements or entries of the sequence are outright, we can use subscript notation. For example, we could say something like “Let \(A=\{a_1,a_2,a_3,a_4,…\}\) be a sequence such that… .” Each of the \(a_1,a_2,\) etc. represent the elements of the sequence, and the subscript number indicate that element’s position in the sequence. So, if given an arbitrary sequence like \(A\) above, the element \(a_{23}\) represents whatever the 23rd element of the sequence \(A\) is.

Generating Sequences from a Closed Formula

The subscript notation above is useful when each value of a given sequence depends on its position in the sequence. For instance, let \(B=\{b_1,b_2,b_3,b_4,b_5,b_6,…\}\) be defined as the sequence

$$B=\{2,4,8,16,32,64,…\}=\{2^1,2^2,2^3,2^4,2^5,2^6,…\}.$$

The first entry is \(b_1=2^1\), the second is \(b_2=2^2\), the third is \(b_3=2^3\), and so on. That is to say, the power on \(2\) was dictated by what entry in the sequence we are talking about (first, second, third, etc.).

Thus, in cases like these, where each value of the sequence depends on its location, we can define a rule that generates the sequence, entry-by-entry. Such a rule is also called a closed formula.

In the case above, we could define the sequence by saying “Let \(B=\{b_1,b_2,b_3,b_4,b_5,…\}\) be the sequence defined by \(b_i=2^i\).” The \(i\) (called the index of the sequence) in this case indicates position in the sequence, and the expression \(b_i=2^i\) tells you that the element at position \(i\) is \(2^i\). For instance, the element in \(B\) at position \(i=12\) is given by \(b_{12}=2^{12}=4096\).

We can use rules such as the above to generate the values of our sequence entry-by-entry.

Another way of thinking about generating sequences like the above is that the rule \(b_i\) gives you a function that helps you compute each entry of the sequence. And, to generate each element of the sequence, all one needs to do is plug in each natural number, \(1,2,3,4,\) etc., one after the other.

To compactify the notation for sequences like \(A=\{a_1,a_2,a_3,a_4,…\}\), we can write instead \(A=\{a_i\}_{i=0}^\infty\) or \((a_i)_{i=1}^\infty\) (or \((a_i)\) for shorthand when the sub and superscripts are known/understood). The subscript \(i=0\) indicates where the sequence starts counting (in this case, we start counting at \(i=1\)) and the superscript indicates where the sequence stops counting (in this case, the \(\infty\) indicates that we don’t stop counting).

Caveat: Starting Points for Sequences

Sequences don’t need to “start counting” at 1. One could define a sequence like \(D=(d_i)_{i=4}^\infty\) where \(d_i=3+4i\). Writing out the elements of this sequence gives us

$$D=(3+4(4),3+4(5),3+4(6), 3+4(7),…)=(19,23, 27,31, …)$$

Notice that the first element was found by starting with \(i=4\) (i.e. plugging in \(4\) instead of \(1\)).

Finding Rules from Sequences Listed Directly

Perhaps in an earlier math class, you were given a sequence like

$$\{2,4,6,8,10,…\}$$

and were then asked to “find the next three terms/entries, assuming the pattern continues.” To do this, you would notice the pattern, and then extrapolate the next three terms of the sequence based on that pattern. In this case, obviously, the next three numbers are \(12,14,16\). One could keep listing out numbers if they wished.

CAVEAT: The key assumption with the above example was that the pattern continued. Without that assumption, technically, there is no way to know what the next element of the sequence is.

But what if instead of being asked to list off the next several elements, someone asked you to find the 1,023,353rd element of the sequence, based on the pattern. You’d kill a few trees worth of paper if you tried to list out everything up to the 1,023,353rd element… and what if you miscounted? Alternatively, you could write a program to do all this for you. All this sounds so painful, considering that the pattern is so obvious!

We could instead just come up with a rule that tells us what the \(i\)th entry of the sequence is, based on the pattern we see.

Recall that for sequences like \(F=\{f_i\}_{i=1}^\infty\) where \(f_i=i^2\) for all \(i\geq 1\), we could generate the entries of \(F\) by simply plugging in each of \(i=1,2,3,4,…\) into the rule \(f_i=i^2\). Each entry of the sequence is therefore obtained by performing the same operations on \(i=1,2,3,4,…\).

Thus, when given a sequence defined by an explicit list, such as \(A=\{a_1,a_2,a_3,a_4,…\}\), we may be able to find a rule \(a_i\) by determining what operations were performed on \(i=1\) to get the first entry \(a_1\), what operations were performed on \(i=2\) to get the second entry \(a_2\), on \(i=3\) for the third entry \(a_3\), and so on.

If the same operations are performed on each of \(i=1,2,3,\) etc. to get \(a_1,a_2,a_3,…\), then that is what gives you your rule.