Chapter+13

=== //**Chapter 13: Sequences, Induction, and the Binomial Theorem**// ===

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= 13.1 Sequences =

Here is a nice video on Factorials:

media type="youtube" key="4j66DS_XTSo" height="344" width="425"



Watch this nice video explaining summation notation:

media type="youtube" key="VvwIxz5GEqc" height="344" width="425" = 13.2 Arithmetic Sequences =  An arithmetic sequence may be defined recursively as,  a1 = a , a n – a n-1 = d where a=a 1 and d are real number. The number **a** is the first term of the sequence, and the number d is called the '**common difference'**
 * An __Arithmetic Sequence__ is a sequence where the difference between successive terms is constant. **

The terms of arithmetic sequence follows this pattern : a, a+d, a+2d, a+3d, a+4d

To determine whether the sequence is arithmetic or not, see if there is a common difference. If there is a common difference the sequence is arithmetic.

For example, let say the sequence is 4, 8, 12, 16,........... the common difference is a <span style="font-family: 'normal','serif'; font-size: 11pt; mso-ansi-language: EN-US; mso-bidi-font-family: 'Times New Roman'; mso-bidi-language: AR-SA; mso-bidi-theme-font: minor-bidi; mso-fareast-font-family: '맑은 고딕'; mso-fareast-language: KO; mso-fareast-theme-font: minor-fareast;">2 = 8 <span style="font-family: 'normal','serif'; font-size: 17pt; mso-ansi-language: EN-US; mso-bidi-font-family: 'Times New Roman'; mso-bidi-language: AR-SA; mso-bidi-theme-font: minor-bidi; mso-fareast-font-family: '맑은 고딕'; mso-fareast-language: KO; mso-fareast-theme-font: minor-fareast;">a <span style="font-family: 'normal','serif'; font-size: 11pt; mso-ansi-language: EN-US; mso-bidi-font-family: 'Times New Roman'; mso-bidi-language: AR-SA; mso-bidi-theme-font: minor-bidi; mso-fareast-font-family: '맑은 고딕'; mso-fareast-language: KO; mso-fareast-theme-font: minor-fareast;">2-1 = <span style="font-family: 'normal','serif'; font-size: 17pt; mso-ansi-language: EN-US; mso-bidi-font-family: 'Times New Roman'; mso-bidi-language: AR-SA; mso-bidi-theme-font: minor-bidi; mso-fareast-font-family: '맑은 고딕'; mso-fareast-language: KO; mso-fareast-theme-font: minor-fareast;">a <span style="font-family: 'normal','serif'; font-size: 11pt; mso-ansi-language: EN-US; mso-bidi-font-family: 'Times New Roman'; mso-bidi-language: AR-SA; mso-bidi-theme-font: minor-bidi; mso-fareast-font-family: '맑은 고딕'; mso-fareast-language: KO; mso-fareast-theme-font: minor-fareast;">1= 4<span style="font-family: Wingdings; mso-ascii-font-family: '맑은 고딕'; mso-ascii-theme-font: minor-latin; mso-char-type: symbol; mso-hansi-font-family: '맑은 고딕'; mso-hansi-theme-font: minor-latin; mso-symbol-font-family: Wingdings; msoasciifontfamily: '맑은 고딕'; msoasciithemefont: minor-latin; msochartype: symbol; msohansifontfamily: '맑은 고딕'; msohansithemefont: minor-latin; msosymbolfontfamily: Wingdings;">à <span style="font-family: 'normal','serif'; font-size: 16pt; mso-ansi-language: EN-US; mso-bidi-font-family: 'Times New Roman'; mso-bidi-language: AR-SA; mso-bidi-theme-font: minor-bidi; mso-fareast-font-family: '맑은 고딕'; mso-fareast-language: KO; mso-fareast-theme-font: minor-fareast;">8 - 4 = __'4__' <span style="font-family: 'normal','serif'; font-size: 14pt; mso-ansi-language: EN-US; mso-bidi-font-family: 'Times New Roman'; mso-bidi-language: AR-SA; mso-bidi-theme-font: minor-bidi; mso-fareast-font-family: '맑은 고딕'; mso-fareast-language: KO; mso-fareast-theme-font: minor-fareast;">so the sequence is __arithmetic__. <span style="font-family: 'normal','serif'; font-size: 14pt;"> Another example is, let say {s <span style="font-family: 'normal','serif'; font-size: 9pt;">n <span style="font-family: 'normal','serif'; font-size: 14pt;">} = {2 <span style="font-family: 'normal','serif'; font-size: 9pt;">n <span style="font-family: 'normal','serif'; font-size: 14pt;">+ 4} find the first term and the common difference of this sequence. First, plug 1 to n, then you get the first term __6__. Then the nth term and (n-1)th terms of the sequence are sn = 2n + 4 <span style="font-family: 'normal','serif'; font-size: 16pt;">s <span style="font-family: 'normal','serif'; font-size: 9pt;">n <span style="font-family: 'normal','serif'; font-size: 14pt;">-1 = 2(n-1) +4 = 2n+2 The difference is s <span style="font-family: 'normal','serif'; font-size: 9pt;">n <span style="font-family: 'normal','serif'; font-size: 14pt;"> – s <span style="font-family: 'normal','serif'; font-size: 9pt;">n <span style="font-family: 'normal','serif'; font-size: 14pt;">-1 = (2n+4) – (2n+2) = 4 – 2 = __2__

<span style="font-family: 'normal','serif'; font-size: 16pt;">a <span style="font-family: 'normal','serif'; mso-bidi-font-size: 10.0pt;">n = <span style="font-family: 'normal','serif'; font-size: 16pt;">a + (n-1)d <span style="font-family: 'normal','serif'; font-size: 14pt;">where a <span style="font-family: 'normal','serif'; mso-bidi-font-size: 10.0pt;">n <span style="font-family: 'normal','serif'; font-size: 14pt;">is nth term, a is first term of sequence, and d is common difference
 * [nth Term of an Arithmetic Sequence]**

Examples: Find the seventh term of the arithmetic sequence: 4, 7, 10, 13, ... First term is a = 4, and common difference is 3. Using the nth term of an Arithmetic Sequence formula, the nth term is <span style="font-family: 'normal','serif'; font-size: 16pt;">a <span style="font-family: 'normal','serif'; mso-bidi-font-size: 10.0pt;">n = <span style="font-family: 'normal','serif'; font-size: 14pt;">4 + (n-1)3 <span style="font-family: 'normal','serif'; font-size: 14pt;">so the seventh term is <span style="font-family: 'normal','serif'; font-size: 16pt;"> a <span style="font-family: 'normal','serif'; mso-bidi-font-size: 10.0pt;">7 = <span style="font-family: 'normal','serif'; font-size: 14pt;">4 + (7-1)3= 22

Finding a Recursive Formula for an Arithmetic Sequence The 9th term of an arithmetic sequence is 63 and the 14th term is 13. Find the 1st term and the common difference. This is what you know so far: <span style="font-family: 'normal','serif'; font-size: 16pt;"><span style="font-family: 'normal','serif'; font-size: 16pt;">a9=a+8d=63 a <span style="font-family: 'normal','serif'; mso-bidi-font-size: 10.0pt;">14 <span style="font-family: 'normal','serif'; font-size: 14pt;">= a + 13d=13 Since you have a system of equation, you can use elimination to solve for d, the common difference. -5d = 50 so __d = -10__ To solve for a, use one of the equations in the system of equations by plugging in d a<span style="font-family: 'normal','serif'; mso-bidi-font-size: 10.0pt;">9 <span style="font-family: 'normal','serif'; font-size: 14pt;">=a+8(-10)=63 <span style="font-family: Wingdings; mso-ascii-font-family: '맑은 고딕'; mso-ascii-theme-font: minor-latin; mso-char-type: symbol; mso-hansi-font-family: '맑은 고딕'; mso-hansi-theme-font: minor-latin; mso-symbol-font-family: Wingdings; msoasciifontfamily: '맑은 고딕'; msoasciithemefont: minor-latin; msochartype: symbol; msohansifontfamily: '맑은 고딕'; msohansithemefont: minor-latin; msosymbolfontfamily: Wingdings;">à <span style="font-family: 'normal','serif'; font-size: 14pt;">a = 143

<span style="font-family: 'normal','serif'; font-size: 16pt; mso-bidi-font-family: Arial;">Sn=(n/2)(a1+an) Sum of n terms is the addition of all the terms of the sequence based on n (which is how many terms there are)
 * [Sum of n Terms of an Arithmetic Sequence]**

Example: Find the sum of nth terms of an arithmetic sequence { 4n +8 }, which is 12 + 16 + 20+ ........ + (4n+8) Sequence (4n+8) has a first term of a=12 and the nth term (4n+8) so to get the sum of nth term Sn = (n/2)( a + a<span style="font-family: 'normal','serif'; mso-bidi-font-size: 10.0pt;">n)= (n/2)[12+(4n+8)] = (n/2)(4n+20)

= 13.3 Geometric Sequences; Geometric Series = = =

-When the ratio of successive terms of a sequence is always the same non-zero number, the sequence is called a **// Geometric Sequence //**, or a **// Geometric Progression //**.

Here are a few examples to show how a Geometric sequence may look:
 * 2, 4, 8, 16, 32, 64,128… **a**= 2 **r**= 2
 * 27, 9, 3, 1/3, 1/9, 1/27… **a**= 27 **r**= 1/3
 * <span style="font-family: Arial,helvetica,sans-serif; font-size: 17px;"> 1, 8, 64, 512, 4096... **a**= 1 **r**= 8

Be prepared to see these types of Geometric sequences in this chapter, and other variations too. You will notice that after the sequence is the variable for the first number in the sequence, “**a**”, and the variable for the common ratio, “**r**”.

A geometric sequence, like it’s arithmetic sequence cousin can too be defined recursively as:

**a** 1 = **a**, **a** n = **ra** n <span style="font-family: Arial,Helvetica,sans-serif; font-size: 12.8pt;">-1

Whenever the first term **a**, and the common ratio **r** are ruled together in a geometric sequence, they will follow this pattern:
 * a, ar , ar ** 2 ** , ar ** 3 ... ** ar n

For more emphasis on classifying the difference between Arithmetic and Geometric sequences, try to guess and highlight what you think might me the correct definition of each sequence: **

Arithmetic: CORRECT!! Geometric: FALSE.........
 * 1,2,3,4,5,6,7,8,9...

3, 6, 12, 24... Arithmetic: FALSE........ Geometric: CORRECT!! **

Arithmetic: CORRECT!! Geometric: FALSE........ ** 1/3, 2/6, 3/9... Geometric: CORRECT!! Finding the ** //** n ** ** th ** ** Term in a Geometric Sequence **
 * a n = 15(n-2)
 * Arithmetic: FALSE........

For the Geometric Sequence an, If you ever wanted to know any particular value in the sequence, it can be easily found with this formula:


 * a n = ar n-1 ** r ≠ 0

For an example on how to solve this type of equation, watch this video (made by myself) media type="youtube" key="jd_XUt0MxTM" height="385" width="640"

If you ever wanted to not only find the nth term in a sequence, but add all of the terms on the way there, you could. The equation to find the sum of n terms i s:
 * Finding the Sum of ** //** n // terms in a Geometric Sequence



For another great example on how to find the sum of n terms in a G. sequence, watch this very interesting and well voiced video: media type="youtube" key="SjOmmIK-loA" height="385" width="640"

//** **Finding the sum of an Infinite Geometric Sequence**

If a sequence has infinite terms, ( // n → ∞ // ), then how would you find the sum of all the terms in the sequence?? To find the sum of and infinite geometric sequence, use the following formula: **// If your heart desires more information on sums of infinite geometric sequences, and I know it does, than watch this extraordinary youtube video: media type="youtube" key="LqT3KxFvR6o" height="385" width="640"

For more teaching and practice problems in section 13.3, this is a helpful site: [|Extra learning] //**

= 13.5 The Binomial Theorem = **Introduction**

== You remember binomials, don't you? You probably also remember how tedious it was to distribute them when multiplying out a binomial raised to an exponent. Especially when the exponent is fairly large and you end up with a polynomial of ridiculous size. It takes forever, right? Well that's where the Binomial Theorem comes in. It gives you a quick way to find out the end polynomial without having to multiply the binomials out. Again, this only works when you're dealing with a single binomial raised to an exponent, denoted by. == == Normally when evaluating you would have to continually use the distributive property and multiply it like normal binomials. There is, in fact, a short-cut we can use to quickly achieve the desired end. First, we must explore Pascal's Triangle.Pascal's Triangle is a triangular arrangement of the binomial coefficients, wherein each number in a row is denoted by the sum of the nearest two values in the above row. We will prove that this works later. The triangle always starts at row 0 where there is a single position, containing the value 1. In the case that one of the values is missing, replace it with a 0 and continue the same pattern. Because of this rule, all of the outside values are 1. First row contains two ones. 1 being the only value above them, the other two values are zeros and they both become one. Second row contains a one, a three, and another one. 0+1=1; 1+1=2; 1+0=1. The first six rows are denoted below. ==
 * Pascal's Triangle**

This short video shows you several other patterns within the triangle as well as illuminating what I already stated, if my explanation was insufficient. media type="youtube" key="Zo2JrPjijHc" height="385" width="640" == As you can see, the pattern continues throughout the triangle, never ending and producing an infinite amount of rows. This is implemented in binomials by replacing the n in the previously denoted binomial base format with the number of the row, the first row being 0. as such, we can find or any such equations. For this we would need to use the 5th row with the values1 5 10 10 5 1. The first coefficient is 1 and n is equal to 5, so the first term in the polynomial is or, if you will allow,. If we continue with the terms the next one will have a coefficient of 5 and the exponent for x will decrease by one while the exponent for a increases by one, leaving us with as our second term. This pattern will continue with the coefficients of 10, 10, 5, and 1 and with exponents changing as I already said, x decreasing and a increasing. This will leave us with a final polynomial of. As you may notice, the exponents for x and a are inversely proportional and will always equal a constant of the initial value of n in. If you didn't quite get that, this video may help you. media type="youtube" key="TeE-ypKj8ZI" height="385" width="480" ==

== Pascal's triangle can also be represented with a special notation and, using a formula, you can use this notation to find any value in Pascal's triangle without having to draw the whole thing to get down to the row you need to use. Each value in Pascal's triangle can be denoted by. (Please ignore the divison line, this is not division I just couldn't find a formula editor that exported to the correct format and could do number columns, there shouldn't be a line, simply a column of 2 numbers, represented by n and j). Can be represented as n taken j at a time or, colloquially, n choose j. To evaluate a set of numbers in this way, we return to a symbol we learned earlier in this chapter and implement it in a simple formula. If n and j are integers and then the value of can be found using the equation. For example, if we had 7 taken 4 at a time, we would plug it into the above equation to get the value. Let's try it, shall we? ==

> > > >.

== Now that you understand how to evaluate that notation, we can return to Pascal's Triangle and show how it can be represented with that notation. The first thing you must know is that is equal to 1. Also, when n is equal to j, is also equal to 1. Additionally, is equal to n, again when n is equal to j. With these values in mind, we can now write the first five rows of Pascal's triangle as          (Sorry about how screwed up it is, this is supposed to be a triangle but wikispaces doesn't like you messing with the arrangement of several pictures, apparently)  ==

[[image:CodeCogsEqn_(19).gif]] [[image:CodeCogsEqn_(21).gif]] [[image:CodeCogsEqn_(22).gif]] [[image:CodeCogsEqn_(23).gif]]


== As you can see, n corresponds to the row, and j corresponds to the value in the n row. For example, is the 5th value in the 8th row given that both rows and values start at 0. Using this, you may find many things. You might be asked something like... First of all, take a second to study the equation. Realize that is within the 5th term of the expansion and n=9 so we know that the term is going to be. As such, the coefficient is just going to be the term with removed. Therefore, the coefficient can be represented by. Now we can evaluate it. == == And that's it! We can now combine all of this knowledge into the final formula for the Binomial Theorem. It should make complete sense at this point. The Binomial Theorem states that, given that x and a are real numbers, for any positive integer for n,  ==

== You could also be asked to find a term in the expansion of a binomial. For that you do the same thing as the previous example but keep the x^n part in there. The only other problems you may encounter are things such as Expand the binomial, or evaluate the expression .==

Practice Problems

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== **5. 6. 7. 8. If you need any more practice problems. [|This site] is good. [|This page] may also help. ** ==