Honors Algebra II/Trig
January 1, 2002
Honors Math Project: Sequences
When first looking at such a problem, one may start to think that they will have to manually add up 203 years worth of interest on a new amount of $ every year! Yet if one knows the laws and equations of mathematical sequences, one can solve this problem in a matter of a few easy steps. Yet prior to solving such a problem, one must first learn the different types of sequences and how to distinguish them from each other according to their differences.
The word sequence in itself suggests it to be a list of members that are organized in a numerical order so that there is a first member (a1), second member (a2), and so on (an). These members are called the terms of the sequence. Like all functions, a sequence can be infinite or finite. In an infinite sequence, the domain is the set of all positive integers, yet in finite sequence the domain is restricted and consists of a certain positive n [number] of integers.
In order to be able to find any term in the sequence, you must first find the expression of the sequence. An expression is an actual pattern that gives you the ability to find any term in the sequence for a corresponding positive member on the number line (1,2,3, etc…). You can find an expression by using just the first couple of given terms in a sequence. For example, if you were given 1,2,4, and 8 as the terms of a sequence, you could then see the obvious pattern to be an = 2n, where n represents any positive number on the number line (1,2,3,4 represent the 1st 2nd and 3rd terms and etc…). You could then use this expression to find any term in the sequence. For example, by knowing the expression of a sequence to be an = 2n, you can find the 4th term easily by plugging in 4 for n and getting the term to be 8. You can do the same if you want with the 450th term, which would be 900 (2 x 450 = 900). From here you can see why sequences are functions; the output is always dependent on the different values plugged in for n. The only hard step lies in trying to find the expression itself. Yet to make it easier, you can line up the given terms (an) and the positive integer to which they correspond on the number line (1st, 2nd, 3rd, 4th, 5th and etc…) underneath each other in order to help you see a pattern with more ease and clarity. Patterns may involve other functions such as multiplication, division, squaring, square rooting and etc. Yet there are some sequences which waiver from this basic method and contain their own special methods of solving.
Recursive sequences and factorials have different rules pertaining to them and look and act differently than the sequences in the previous paragraph. In a recursive sequence, the terms are defined using previously derived terms. Thus, to define a sequence recursively, you will need to be told 2 or more terms at the start as a given. From there you can derive the following terms with the general formula ak = ak-2 + ak-1 where k is greater than or equal to 2 (this is why you will need 2 given terms to be able to use the recursive formula). Yet up till now you’ve seen regular terms expressed in regular number notation. Yet there is a very strange type of term that is defined with products called factorials. Factorials are defined as n! = 1 x 2 x 3 x 4… x n (o is special for o! = 1). For example, if you are asked to find the 4th term of the sequence an = 2/n!, you would substitute 4 for n and find 4! (24) and get the term to be 2/24 or 1/12. Because factorials use multiplication, when dividing factorials, you can apply the rules of cancellation and
One of the two most important types of sequences, is the arithmetic sequence, which can be recognized if the difference between any 2 consecutive terms in a sequence is the same. For example, the sequence of 3, 8, 13, 18, 23 has what is called a common difference of 5, which is the difference…