CLEP College Algebra/Sequences and Series

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In mathematics, it is important to find patterns. That is what mathematicians do almost everyday of their lives. How they determine patterns is different depending on the type of mathematics they work with. For college algebra, determining patterns is part of the curriculum. The problem below demonstrates one way we determine these patterns. Keep in mind that the problem below is an Exploration and most likely not representative of the types of problems you may see in the CLEP College Algebra exam.

Exploration 1: Imagine the Standard English Alphabet on a page, listed in order from left to right. After the last letter in the Standard English Alphabet, Z, define a next letter AA that will be added to the Standard English Alphabet. Continue listing the alphabet as normal. Assume letter A is defined as the first letter in this "New Alphabet." At what position will you find the letter HY in the "New Alphabet."

In the standard English Alphabet, there are letters in total. Let us define the list of all the letters A-Z as the "Old Alphabet."

Adding another to A will give us a new letter, . Adding this to the "Old Alphabet" obviously does not make it the old set anymore, so let us define the alphabet in which you add new letters to the old one the "New Alphabet" How many of the new letters are in this set? Well, each letter from "Old Alphabet" must get another additional letter to make the "New Alphabet" Therefore, for each letter, letters are added to the old one. Since each letter is used, exactly are created in the "New Alphabet" The figure below may help demonstrate this new fact.

The left brace tells us how many letters are in "Old Alphabet"; the top brace for each "row" tells us how many new letters are created per letter of the alphabet. Since 26 letters are created for each letter of the "Old Alphabet," add every new letter created into the "New Alphabet." Since there are rows, and each row creates new letters,

Given that we know the total number of letters in the "New Alphabet," we can find out what position letter HY is in this "New Alphabet." First, the letter position of H is . Second, the letter position of Y is . Since the above figure shows a table, we can find HY by looking up that "coordinate." Since the position of HY contains all the terms in that given area, we can multiply the two values to get the area. This is not the final answer, however. Remember we excluded some 7 other letters by multiplying in that block; ergo:

What were we doing in the problem above? Essentially, we were simply trying to find what position some "term" is in. Does it not intrigue you to see math try to find a position of some "term" in a list? We have problems like these as mathematicians because the patterns underlying a "list" of numbers can help us determine new facts of mathematics. After all, what were we doing when using functions? We were trying to find a number using a pattern (the function). Unlike the previous sections, however, we were not given a formula. Luckily enough, it is not difficult to make a formula for a given "list" of numbers. Before we dive into these new problems. It helps to establish definitions.

Definition.
A sequence is a list of elements, such as numbers, figures, or letters, that is generally written in some pattern.
Objects in Sequences.
A term is an item found within a sequence.

There is a little disclaimer to get out of the way before we try to solve some problems. First, a sequence can have no pattern. However, for our purposes, we will not count any list of numbers in which no pattern exists. Second, even in simple sequences, numbers of any kind can be named if they follow after a sequence rule. For example, here's a sequence in which the rule is to list prime numbers: If you were like most people, you would probably name the prime numbers in order. However, you could perhaps finish the sequence like this: For our purpose of standardization, we will follow a pattern by stating what number you must find first or identify the pattern.

Let's begin exploring the world that is Sequences and Series.

Sequences[edit]

As you already know, a sequence is a list of objects that generally follow a pattern. However, the type of pattern that is described will classify sequences into either arithmetic sequences or geometric sequences. Each will be explored in depth within the next sections.

Arithmetic Sequences[edit]

Location of a term.
An index is the location of a term within a sequence, usually denoted by or .
Definition of Arithmetic Sequence.
An arithmetic sequence is a sequence in which an added real number , called the difference, is added to each successive term, except the first term, , such that the sequence forms in a one-to-one correspondence.

An example could perhaps help you figure with the formal definition above: The sequence has a one-to-one correspondence with the general sequence since the first term , second term , third term , and so on. The difference is the amount added to each previous term to get the new term. For , or . Solving for is the difference of the two terms. In this example, the difference is . This is how we define an arithmetic sequence.

Recursive Formula[edit]

Often times, we want to generate a sequence using a formula (we are mathematicians, after all, and we like to study sequences to see if there are any general patterns). If we want to find , we may use the following formula:

However, the above formula could describe any sequence that has that general pattern. To fix this, we need to describe the first term as well when using the formula above. There are two ways to describe this formula:

  1. We have it horizontally deliniated: .
  2. We have it vertically deliniated:

To save space, we will horizontally deliniate formulas for arithmetic sequences in this WikiBook.

Formulas in which the first term is identified along with an equation in which the previous term is added by to get the next term is called a recursive formula.

A recursive formula is a formula that describes how to get a term or many terms of a sequence by stating the starting value or values and each previous term or terms and adding difference to each subsequent term.
Recursive Formula for an Arithmetic Sequence

Given an initial value and constant difference , the th term in an arithmetic sequence is given by the previous th term of the sequence:

OR
Example 1.1.1.a: Find the th term to the arithmetic sequence

To determine the next term, we first need to find the difference . Note that an arithmetic sequence will have the next term add to the previous term. Since that is how the arithmetic sequence works, is a valid way to find the difference between two terms. Solving for , we find the constant difference is . Since the difference is the same for each given term in the sequence, we can find the 7th term by adding to , which gives us . Finally, add to the next term to get the final answer:

There are many reasons why it is more important to have a recursive formula. It is not always slow; it may be easier to understand. The next example shows why this is exactly true

Example 1.1.1.b: Write the recursive formula to the sequence .

Many of you will perhaps know this famous pattern as the Fibonacci Sequence. For those of you who do not know this sequence, the way we determine the next term is by using the previous terms and adding them together. In our notation, we would say that the term at index , is equivalent to . Remember, however, we are not done. If a mathematician saw the sequence , he (or she) would determine that also describes that sequence. Therefore, we must list the first two terms because listing only the first term would not allow us to get the next term. This means our final answer is

Note that Example 1.1.b is not an example of an arithmetic sequence. Your next exploration will be to determine why this is true. Along with that, you will use your critical thinking skills to argue for or against something in the explorations after that one.

Exploration 2: Explain why the Fibonacci Sequence is not considered arithmetic. Determine how to change the formula for the Fibonacci Sequence so that it becomes an arithmetic recursive formula; explain why must the formula be written that way.

A term at index is not added by a common difference , where must remain constant. By definition, the Fibonacci Sequence does not have a constant difference to get to the next term because the formula does not have a constant difference. Ergo, the Fibonacci Sequence formula is not arithmetic, but it is recursive. To make it arithmetic, identify only the first term because we can find the second term by using the constant difference , then add the previous term of , , by constant difference : .

Exploration 3: Let the first two terms of a sequence be and where .
  • Argue either for OR against the claim that the recursive formula is arithmetic and recursive.
  • If you disagree that is arithmetic and recursive, write a formula that is arithmetic and recursive.
  • If you agree that is arithmetic and recursive, explain why it is so.

Finally, if , , and ,

  • argue either for OR against the idea that the formula is arithmetic and recursive.
  • The sequence generated by using the formula is For a sequence to be arithmetic, the common difference must be constant. While the common difference is constant, the sequence must be in the form ; otherwise, it is not arithmetic. Therefore, the recursive formula is not arithmetic. To make it an arithmetic recursive formula, change the previous formula such that .
  • If , then the sequence generated by using the formula is However, for a sequence to be arithmetic, each term must be added by the constant difference. Since the term in index is not added by , the formula is not arithmetic.
  • Exploration 4: Let the first term . Argue either for OR against the idea that sequence is arithmetic if you change it to

    Since is the arithmetic recursive formula, is valid. The only way for to be true is by making . However, since you are adding each term by using a constant difference , and you are using the previous term to do so, the new sequence forms: By definition, the sequence formed is arithmetic.

    Direct Formula[edit]

    By now, you may be wondering if there is a way to find the term of an arithmetic sequence directly. Well, there is. Before giving you the formula, let us go through the motions for our general recursive arithmetic formula . Let's chart the recursive arithmetic formula.

    If you think about it, the table above is basically a linear function, although starting at instead of . Write out the function as . We get near our answer. Our independent variable is horizontally translated by to the right, so is our function. In fact, we found our direct relationship. Rewrite it the way we normally write it and we found our direct formula: .

    A direct formula describes how to get a term or many terms of a sequence by only using a formula that directly finds a particular value, without stating the first term or terms of a sequence.
    Direct Formula for an Arithmetic Sequence

    Given an initial value and constant difference , the th term in an arithmetic sequence is given by the direct formula:

    Example 1.1.2.a: An arithmetic sequence is discovered. What is the th term in the sequence?

    As always, before we can determine the answer, we need to find the "rate of change" of our sequence. Since , we know that . The second term is , so when , . Of course, now we can solve for :

    ;
    ;
    .

    Since we now know the common difference, we can find the smallest th term in the sequence. By using our direct formula for an arithmetic sequence, we can find the index in which it is possible. Since , we find the th term by substituting . Ergo, . Solve for to get the final answer.

    ;
    ;
    ;

    The example above would be a routine, straightforward problem in the CLEP College Algebra exam. However, as practice makes perfect, we will also have non-routine problems that involve thorough understanding of the topic and concepts and skills learned, which will make up 50% of the exam. This is why it is important to do the explorations. While they may not be on a CLEP exam, they are vital in making you think like a mathematician. The next problem will be non-routine problem.

    Example 1.1.2.b: What is the smallest index needed to find a negative term in the arithmetic sequence ?

    As always, before we can determine the answer, we need to find the "rate of change" of our sequence. Since , we know that . We know that the second term is , so when , . Of course, now we can solve for :

    ;
    ;
    .

    Since we now know the common difference, we can find the smallest term needed to reach a negative number in our sequence. By using our direct formula for an arithmetic sequence, we can find the index by solving for . Since , we find the minimal index in which it is possible to have a number less than zero. Ergo, . Now all we have to do is solve for .

    ;
    ;
    ;
    ;
    .

    Since index must be greater than , the minimal index required to find the term that is negative is at

    Geometric Sequence[edit]

    Definition of Geometric Sequence.
    An arithmetic sequence is a sequence in which a multiplied real number , called the common ratio, is multiplied to each successive term, except the first term, , such that the sequence forms

    As always, if you are unable to understand, try a few examples of numbers to think of in your head. Let the common ratio and let . The next term , so . If you keep the pattern going for each term of the sequence, you would get the following:

    Recursive Formula[edit]

    As with the arithmetic formula, you can find the recursive formula and the direct formula for a geometric sequence. Since every term is multiplied by common ratio , with the first term , for any term of index , to find the next term requires knowing the previous term. Ergo,

    Recursive Formula for a Geometric Sequence

    Given an initial value and constant difference , the th term in an arithmetic sequence is given by the previous th term of the sequence:

    OR

    Direct Formula[edit]

    By now, you may be wondering if there is a way to find the term of an arithmetic sequence directly. Well, there is. Before giving you the formula, let us go through the motions for our general recursive arithmetic formula . Let's chart the recursive arithmetic formula.

    If you think about it, the table above is basically an exponential function, , starting at . Write out the function as . We get near our answer. Our independent variable is made lesser by , so is our function. In fact, we found our direct relationship. Rewrite it the way we normally write it and we found our direct formula: .

    Direct Formula for an Arithmetic Sequence

    Given an initial value and common ratio , the th term in an geometric sequence is given by the direct formula:

    Example 1.2.2.a: A geometric sequence is discovered. What is the th term in the sequence?

    To find the answer, we need to know the common ratio of the sequence above. Pick any arbitrary term in the sequence and apply it to the direct geometric formula: . Find :

    Knowing the value of , you can find the th term in the sequence by using the direct formula:

    As always, these examples are things you can work through yourself or follow along so that you can see how to do a problem.

    Exploration 5: Let the first term . Argue either for OR against the idea that sequence is geometric.

    A geometric sequence must have the terms be multiplied by a common ratio . If the sequence were arithmetic, a term must be added such that adding the previous term by a constant difference will result in the alternation of negative and positive terms. Since that is impossible, you must multiply the terms in the sequence above by a common ratio .

    Exploration 6: A power function is a function that has the power term be any real number or , where . Let and . If ,
    • multiply each value of by a common ratio . Determine whether the range will also be a geometric sequence through example using any value of .
    • prove that multiplying the domain, , by a common ratio will give a range that is geometric (where the common difference for the domain is .
  • Since the domain of must be a geometric sequence multiplied by a common ratio , multiply each value of by . The following geometric sequence is formed: Applying those values to the function , the following is the sequence generated: Each sequence must have values multiplied by . As such, the range of must be geometric.
  • The question is asking whether or not will give a range for such that you multiply by a constant. Let's define that constant as . Let . Since , if , then . Since , function . Since any term will give , function . Since , function . Since and are constant, the expression will be constant. Since you are multiplying the range, , by a common ratio (otherwise known as a geometric sequence), given the constant , then when multiplying the values of the domain, the range must be multiplied by a constant:
  • Sigma Notation[edit]

    Series[edit]

    Arithmetic Series[edit]

    Geometric Series[edit]

    Infinite Geometric Series[edit]