7-1

Review of 7-1

This week I'm going to do a review on lesson 7-1 since we've been doing trig review all week. In this lesson, we learn how to convert degrees to radians, radians to degrees, degrees to minutes and seconds, and minutes and seconds to degrees. We also learned how to find co-terminal angles.

- To convert degrees to radians: degrees x pi/180 (In the calc, you type degrees/180, and convert to a fraction, then put pi beside it.

- To convert radians to degrees: radians x 180/pi (Pi will cancel out)

- To covert degrees to minutes and seconds: For minutes, you take what is behind the decimal and multiply by 60. Once you get that answer, you take what is behind the decimal and multiply by 60 which will give you your seconds.

- To convert from minutes and seconds to degrees: degrees + min/60 + sec/3600

- To find a co-terminal angle: n +/- 360 degrees

Example 1:

a) convert 315 degrees to radians: 315 degrees x pi/180

= 7pi/4

b) convert -pi/2 to degrees: -pi/2 x 180/pi

= -90 degrees

c) convert 1.32 to degrees, minutes, and seconds: 1.32 x 180/pi

= 75.630 degrees

- .630 x 60 = 37.8 minutes

- .8 x 60 = 48 seconds

= 75 degrees, 37 minutes, 48 seconds

Example 2:

a) Find a positive co-terminal angle to -100 degrees: -100 + 360

= 260 degrees

b) Find a negative co-terminal angle to 500 degrees: 500 - 360(2)

= -220 degrees

Review of Probability

This is simple if you remember the difference between mutually exclusive and not mutually exclusive.

Notes:

If two events are mutually exclusive, then they cannot happen at the same time.

P(A or B) = P(A) + P(B) - P(A & B).

-If the events are not mutually exclusive, then P(A or B) = P(A) + P(B).

Ex 1: A standard deck of cards consists of 52 cards, with 13 cards in each of four suits (clubs, spades, diamonds, and hearts). Clubs and spades are black, and diamonds and hearts are red. The face cards consist of jacks, queens, and kings. Assuming the deck is shuffled properly, what would be the probability that a card is:

• red = 26/52 = 1/2
• A black club = 13/52 =1/4
• blue = 0/52 = 0
• A jack = 4/52 = 1/13

-Sameer

review of matrices,WOOHOO

This week, since we just reviewed trig and took our exam, I’ve decided that I would review the wonderful world of matrices with all of you. So here we go with the blogging and what not.

-A^t means transpose. That simply means switch the rows and columns.
-To add matrices they must be the same dimensions (number of rows and columns)
-To complete scalar multiplication you multiply every entry by the number.

EXAMPlE TIIIIIIIIIIIIME
M= [7 1 5]
[4 3 2]

A. What are the dimensions of the matrix?
2x3 ( There are two rows and three columns )

B. Find m^t
[4 7]
[3 1]
[2 5]

Example TWO:

Add the matrix from the last example and the matrix
[8 3 4]
[6 2 9]

Simply add the corresponding spots.
[7+8 1+3 5+4]
[4+6 3+2 2+9]

The answer would then be:
[15 4 9]
[10 5 11]

So that is how you work with matrices.

--Sarah (:

MORE BLOGS YAY

Lets talk about solving exponents as a variable

1) take the log or ln of both sides

*use ln where there is an e*

2) put into calculator if asked exact values don't get input in a
calculator

EX:
4^x=16
log4^x=log 16
xlog4=log16
x= log16/log4
x=2

BLOGS YAY

cory rules

exponents that are variables

today im showing you how to solve for exponents that is a vairable. the steps are listed below and will be followed with examples to clarify the process.
1) take the log or ln of both sides *use ln where there is an e*
2) put into calculator if asked exact values don't get input in a calculator



EX:
3^x = 12
log 3x = log 12
x= log 12 / log 3
x= 2.26

(the square root of) e^x = 50
ln e^1/2x = ln 50
x= (ln 50 / ln e) *2
x=7.82

Cos(a +/- B) and sin(a +/- B)

Things you need to know:

• cos(a +/- B) = cosa(cosB) -/+ sina(sinB)
• sin(a +/- B) = sina(cosB) +/- cosa(sinB)
• sinx + siny = 2sin(x + y/2) cos(x - y/2)
• sinx - siny = 2cos(x + y/2) sin(x - y/2)
• cosx + cosy = 2cos(x + y/2) cos(x - y/2)
• cosx - cosy = -2sin(x + y/2) sin(x - y/2)

Example 1: Find the exact value of cos75 degrees.

• You will first go to your trig chart to find out what adds to give you 75 for cos.
• Those numbers are 45 and 30.
• You are going to use those numbers in the first formula from above.
• cos(45 + 30) = cos45(cos30) - sin45(sin45)
• Now you plug in what those are from the trig chart.
• (square root of 2/2)(square root of 3/2) - (1/2)(square root of 2/2)
• square root of 6/4 - square root of 2/4
• Final answer: square root of 6 - square root of 2/4

Example 2: sin30(cos15) + cos30(sin15)

• For this problem they give you the two numbers they are adding so you have to replace it with one of the formulas.
• If you look at the problem and the formulas you have from above, it is the same as sin(a + B).
• Now you can replace.
• sin(30 + 15) = sin(45)
• Now that you have gotten an answer, you have to see what sin(45) is on the trig chart.
• sin(45) on the trig chart is square root of 2/2.
• Final answer: square root of 2/2.

-Amber :)

Review of Trig Identities

This weekend I am going to review with you all the information that we learned about identities. We learned this in trig. The identities are used to solve trig function equation. These are things like sin, cos, and tan. There are some identities that you need to know for this section.

Notes:

Reciprocal
• csc (theta)=1/sin (theta)
• sec (theta)=1/cos (theta)
• cot (theta)=1/tan (theta)
• tan (theta)=sin (theta)/ cos (theta)
• tan (theta)=cos (theta)/ sin (theta)

Pythagorean
• sin^2 (theta) + cos^2(theta) = 1
• 1 + tan^2 (theta) = sec^2 (theta)
• 1 + cot^2 (theta) = csc^2 (theta)

Cofunction
• sin(theta) = cos (90 degrees-theta)
• cos (theta) = sin (90 degrees-theta)
• tan(theta) = cot(90 degrees-theta)
• cot (theta) = tan (90 degrees-theta)
• sec (theta) = csc (90 degrees-theta)
• csc(theta) = sec (90 degrees-theta)

Example: Simplify: (sec^2x – tan^2x)/cos x
• You would use the identity that contains tan^2 and sec^2
• That would give you 1/cos x
• Your final answer would be secx


-Braxton-

Trig Reviewww

This week, I'm going to do some review from one of the trigonometry sections. If you do remember any trig, you should be good to go and could probably work my example right now. But incase you don’t remember everything, i'm going to give you some information. all you need to know right now in trig is basic formulas and the trig chart. For starters, you must remember:

• Tan=sin/cos
• Sin=1/csc
• Sec=1/cos
• Cos=1/sec
• Tan=1/cot
• Cot=1/tan
• Csc=1/sin

Apart from knowing that, you must also know the Pythagorean Identities. To refresh your memories, these are the Identities you will need to use when doing a lot of trigonometry.

• 1+tan^2=sec^2
• 1+cot^2=csc^2
• Sin^2+cos^2=1

Now that we’ve went over some of the basic things needed to work the examples I am about to show you, we can begin!!

Example 1: Simplify:

(Sinx^2+Cosx^2)/ Sinx

1. Look for identities. You have one, so simplify it.
2. You now get 1/Sinx. You can simplify this by referring to the notes above.
3. You get Cscx as your final answer J

And there's a trig review!

Carleyy

Using formulas cos(a +- B) and sin (a +- B)

This week I am going to review how to work problems using formulas for cos(a+ or
- B) and sin(a+ or - B). There are 6 different formulas for these types of problems. You should already know these formulas because I have already went over these types of problems, but to refresh you memory I will list them again.

The formulas are:

• cos(a + or - B)=cosa cosB - or + sina sinB
• sin(a + or - B)=sina cosB + or - cosa sinB
• sinx+siny=2sin (x+y/2) cos (x-y/2)
• sinx-siny=2cos (x+y/2) sin (x-y/2)
• cosx+cosy=2cos (x+y/2) cos (x-y/2)
• cosx-cosy=-2sin (x+y/2) sin (x-y/2)
  

Now that you should remember the formulas, I am going to work a few examples which should help you to understand how to work these problems. They are very simple as long as you know the formulas above.

Example 1: cos 105 degrees

• You are going to use your trig chart to help work this problem.
• You are going to use formula 1 to solve this problem.
• Since 45 and 60 degrees are on the trig chart and they add up to equal 105, you are going to use those two degrees.
• cos(45+60)=cos45 cos60-sin45 sin60
• You then plug those into the trig chart.
• You answer is going to be: square root of 2 - square root of 6/4
  

Example 2: sin75 cos15 + cos75 sin15

• You are going to replace this with one of your formulas above.
• That formula above is the same as sin(a+B)
• Once you replace with that formula you are going to get sin(75+15)=90 degrees
• sin 90 degrees on the trig chart equal 1
• 1 is goin to be your answer
  

And that is how you work problems using formuals for cos(a+B) and sin(a+B). Well that is is for this weeek. Byeeeee.

--Halie!

5-7

5-7 Exponential Equations

In this lesson, we learned how to solve for an exponent that is a variable. There are two rules:

1. Take the log or ln of both sides. Use ln when the number is e.
2. Put into your calculator only if asked to. Exact values do not get input into a calculator.

Example 1: Solve the equation by using logarithms.

a) 4^x = 20 (take the log of each side of the equation)

= xlog4 = log20 (divide log20 by log4)

= 2.16

b) 25^x = 2 (take the log of each side)

= xlog25 = log2 (divide log2 by log25)

= .22

Example 2: Solve the equation by using logarithms.

- 4^x = 16(sq root of 2) (take the log of each side)

= xlog4 = log16(2)^1/2 (multiply 16 by 2^1/2)

= xlog4 = log22.63 (divide log22.63 by log4)

= 2.25

Logsssss!

The logarithmic function logb^x and the exponential function bx are inverse of each other, so that means

y = logb^x is equivalent to x = b^y

where b is the common base of the exponential and the logarithm.

The forms above helps in solving logarithmic and exponential functions and needs a deep understanding. Examples, of how the above relationship between the logarithm and exponential may be used to transform expressions, are presented below.

Example 1 : Change each logarithmic expression to an exponential expression.

1. log3^27 = 3

2. log36^6 = 1 / 2

3. log2^(1 / 8) = -3

4. log8^2 = 1 / 3

Solution to Example 1:

1. The logarithmic form log3^27 = 3 is equivalent to the exponential form

27 = 33


2. The logarithmic form log36^6 = 1 / 2 is equivalent to the exponential form
6 = 361/2


3. log2^(1 / 8) = -3 in exponential form is given by

1 / 8 = 2-3


4. log8^2 = 1 / 3 in exponential form is given by

2 = 81/3

logs

today im going to teach you how to put logs in exponential form, expand logs, and condense them. this is a really simple concept once you know what you are doing. its pretty hard to explain any concept unless ther eis an example so lets do some examples.


EX:put in exponential form
log base 4   16 = 2
you take the base which is 4 and you raise is to what the whole thing equals
4^2 = 16

expand:
log (MN)^2
when you expand logs you put in terms of addition if they are multiplied and subtraction if they are divided and you put exponents in front of the equation
2 log M + 2 log N

condense:
log 8 + log 5 - log 4
to condense you just do the expanding steps backwards.
log (8)(5) / 4
log 40/4
log 10
since this is log 10 the base i also 10 which cancles everything and your equation equals 1

Blog STuff YAY

LETS TALK ABOUT LOGS!!! YAY!!!!

Notes from class

• Logs can be written as exponents and it works the other way


• Convert to exponential form to solve a log


• If no base is there than it is 10


• For natural log, no base is implied to be e


• Shortcut: If b and x are equal, they cancel and leave you with exponent

Example

solve the log

log(base 2) 4 = x

2^x=4

x=2

YAY!

Logs

I am going to show you all the process of solving problems that include logarithms. This process is quite simple if one knows how to look out for key components in the logarithms.

Notes:
• One thing to remember is if the base of the log isn't there then it is assumed that the base is 10.
• Another thing is logs can be written as exponents and vice versa for example loga=10, it is implied that log base 10 raised to the tenth power is a.

• For natural logs not having a base then it is assumed that the base is e.
• here's a shortcut: If b and x are equal, then they cancel out and the exponent left will be your answer

Ex 1: log (base 3) 81=x

• 3^x=81
• When solving for x you get 4 since 3^4 = 81
• Your final answer will be 4

-Sameer

We spent the past week learning about logarithms and all the wonders that come with them and by golly it was exciting! This blog is all about those logarithms.

Alright, so here are the things you will need to know:

• Logs can be written as exponents and vice versa. For example: log x = a could also be written as 10^a = x. 10^c = 4 could also be written as log 4 = c
• Whenever you solve a log, you have to convert it to exponential form
• If the base of the log is not written, it is implied that the base is 10
• If the base of the natural log is not written, it is implied that the base is e
• Shortcut: If b and x are equal, then they cancel out and leave you with the exponent as your answer

EXAMPLE:
Rewrite the following in exponential form then solve.

Log28=3

Remember, base^right side of equal sign=left side of equal sign

2^3=8

THAT IS ALL.
--Sarah(:

We spent the past week learning about logarithms and all the wonders that come with them and by golly it was exciting! This blog is all about those logarithms.

Alright, so here are the things you will need to know:

• Logs can be written as exponents and vice versa. For example: log x = a could also be written as 10^a = x. 10^c = 4 could also be written as log 4 = c
• Whenever you solve a log, you have to convert it to exponential form
• If the base of the log is not written, it is implied that the base is 10
• If the base of the natural log is not written, it is implied that the base is e
• Shortcut: If b and x are equal, then they cancel out and leave you with the exponent as your answer

EXAMPLE:
Rewrite the following in exponential form then solve.

Log28=3

Remember, base^right side of equal sign=left side of equal sign

2^3=8

THAT IS ALL.
--Sarah(:

5-3

This weekend I am going to teach you all how to work problems that involve logarithms. Before I show you any examples, there are a few things that you need to know.

Notes:
• Logs can be written as exponents and vice versa. For example: log x = a could also be written as 10^a = x. 10^c = 4 could also be written as log 4 = c
• Whenever you solve a log, you have to convert it to exponential form
• If the base of the log is not written, it is implied that the base is 10
• If the base of the natural log is not written, it is implied that the base is e
• Shortcut: If b and x are equal, then they cancel out and leave you with the exponent as your answer

Example: log 1000 = x
• 10^x = 1000
• When you solve for x you get 3 because 10^3 = 1000
• Your final answer would be 3


-Braxton-

5-3 Logarithms

Okay, so this week I am going to explain how to work logarithm problems. This is very simple, although there are a lot of different ways to solve these problems. There are also many different problems that contain logarithms. I am only going to show you a few though. First I am going to give you a few notes that are going to help you along the way.

Notes:

• Logs are another way to write exponents. For example, logb X=a as an exponent is b^a=x/
• To solve a log, you write as an exponent.
• If no base is written it's implied to be 10.
• The base of ln is always e.
• If b and x are equal they cancel along with the log, leaving the exponent as your answer.

Okay, so now that you know a few basic things about logarithms, I am going to work a few example for you.

Example 1: log 100 = x

• Since there is no base number, you automatically assume that the base is 10.
• You then switch it around and get 10^x=100
• Once you do that, you solve for x and you end up with 2.

Example 2: ln x=-1.5

• Since there is no base number, you automatically assume that the base is e.
• You should know that because of the notes I gave you above.
• You then switch is around just like you did in example one and get e^-1.5=x
• You then have to solve for x and you should end up with x=.22 as your answer.

And that is how you work these types of problems. They are very simple and should be easy to learn. Well that is it for this weeek. byeee.

--Halie!

5-6

5-6 Laws of Logarithms

In this lesson, we learned how to prove and use laws of logarithms.

-If M and N are positive real numbers and b is a positive number other than 1, then:

1. logb MN = logb M + logb N
2. logb M/N = logb M - logb N
3. logb M = logb N if and only M = N
4. logb MO^k = klogb M, for any real number k

Example 1: Write the expression in terms of log M and log N.

a) log (MN)^2

= 2 (logM + logN)

b) log 1/M

= -log M

Example 2: Write the expression as a rational number or as a single logarithm.

a) log 8 + log 5 - log 4

= log 8(5)/4

=log 10 (There is no base, so it's automatically a base of 10, which makes the bases the same. - Therefore, the bases cancel out)

= 1

b) 2 ln6 - ln 3

= ln 36 - ln 3

= ln 12

Finding the infinite sum of geometric series is quite simple. Simply applying the formula will get the answer. Remember that this will only work for geometric series not arithmetic series.

This will only work if the infinite sum of a geometric series if r < 1
The formula for finding the infinite sum of a geometric series is:
S= ((t1) / (1-r))
To find where an infinite geometric converges, set r < 1 and solve for the x term
this how you are to write a repeating decimal as a fraction. To do this follow this formula: (what’s repeating / place-1)

Ex 1: Find the sum of the infinite geometric series: -3, 12, -36, 144...
• S=t1 / 1-r = -3 / 1- (-4)
• S=24 / 5
-Sameer

here i am trying to catch up on these blogs lol

Infinite Sums
Notes
• You can only find the infinite sum of a geometric series, it will not work for an arithmetic series
• You can only find the infinite sum of a geometric series if r < 1
• The formula for finding the infinite sum of a geometric series is as follows:S= ((t1) / (1-r))
• To find where an infinite geometric converges, set r < 1 and solve for x
• how to write a repeating decimal as a fraction. To do this follow this formula: (whats repeating / place-1)

Example

Find the sum of the infinite geometric series.
6 + 12 + 24…
r= 12/6= 2 24/12=2

solve
6/1-2=6/-1= -6

YAY!! i did a blog

13-5

Alright this week I’m going to talk about section 13-5. It’s all about infinite sums.

First things first, let’s see those notes!
• You can only find the infinite sum of a geometric series, it will not work for an arithmetic series
• You can only find the infinite sum of a geometric series if r < 1
• The formula for finding the infinite sum of a geometric series is as follows:
S= ((t1) / (1-r))
• To find where an infinite geometric converges, set r < 1 and solve for x
• Another thing that you need to know about this section is how to write a repeating decimal as a fraction. To do this follow this formula: (what’s repeating / place-1)

Alright how about an example?

Find the sum of the infinite geometric series.
9-6+4-…
First find r. -6/9= -2/3, 4/-6= -2/3
Next plug into the formula
S= 9 (first term)/1- (-2/3) (that’s r)
S= 9/ 5/3= 27/5

That’s all there is to it guys.
--Sarah

13-5

This week we went over 13-5, which is sums of infinite series. You can never find an infinite sum of an arithmetic series,only a geometric series. You are only able to find a infinite series of a geometric series if |r|<1. The formula for finding the sum of a geometric series is s = (t1/r-1). To find where an infinite geometric series converges set |r|<1 and solve for x. To write a repating decimal, you use the formula as follows: what's repeating/place-1.

Example 1:

find the sum of the infinite geometric series

9-6+4-...

1. first, you have to plug into the formula.

s= 9/1-(-2/3)

2. now solve for s.

s= 9/(5/30

= 27/5

The sum of the infinite geometric series is 27/5.

Example 2:

express the following decimal as a rational number.

0.7777..

1. first, you have to plug into the formula

.7/1-.1

2. solve

= /9

the rational number is 7/9.

Exponents

Exponents are... annoying and I'm not all that great with them BUT this is the first blog I've done in... uh.. a while so bear with me!

There are several rules to exponents that are not necessarily difficult if you do them in the proper order. I'm going to work the following problem to give you an idea of how many different steps a problem may take to simplify.

(3^-2 + 3^-3)^-1 (Order of Operations states to deal with the stuff inside the () first)
(1/6 + 1/9)^-1 (Because the exponents were negative I took the recipricol and dealt with the exponent in the denomenator. After that add the fractions.)
(5/18)^-1 (Because there is a negative exponent on the outside, of the parenthesis you will take the recipricol of the fraction.)
(18/5) = Answer

~ Parrish

13-5

Things you should know:

• You can not find an infinite sum for an arithmetic series.
• Geometric series where |r| < 1 are the only series that have an infinite sum.
• Formula: S = t1/1-r
• Set |r| < 1 and solve for x when you are finding infinite geometric converges.
• When you are writing a repeating decimal as a fraction you do this formula: repeating/place - 1.

Example 1: What is the fraction form of this decimal? .36363636

• First you have to find the number that is repeating: 36
• Now you follow the formula: repeating/place -1
• 36/100-1 (you put 100 because the first 36 is in the 100th place)
• You would then end up with 36/99
• Final answer: 36/99

Example 2: What is the sum of the infinite geometric series: 36 - 12 + 4 - . .

• First you must know that from the formula in the above notes, S stands for the sum.
• T1 is the first term that you are given in the series which is 36
• R is the number that is being multiplied to get to the next number which is -3. It is- 3 because 36/-12 = 3 and -12/4 = -3.
• Now you can plug the numbers into the formula.
• Formula: S = t1/1 - r.
• S = 36/1 - (.3) = 36/4 = 9
• Final answer: S = 9

- Amber :)

13-5

This weekend I am going to teach you all how to work problems that require you to find infinite sums of geometric series. Before I show you all an example, there are a few things that you need to know about infinite sums.

Notes:
• You can only find the infinite sum of a geometric series, it will not work for an arithmetic series
• You can only find the infinite sum of a geometric series if r < 1
• The formula for finding the infinite sum of a geometric series is as follows:
S= ((t1) / (1-r))
• To find where an infinite geometric converges, set r < 1 and solve for x
• Another thing that you need to know about this section is how to write a repeating decimal as a fraction. To do this follow this formula: (what’s repeating / place-1)

Example: Find the sum of the infinite geometric series: 24-12+6-3
• S=t1 / 1-r = 24 / 1- (-1/2)
• S=24 / (3/2)
• S=16


-Braxton-

exponents

this week we did exponents so today im going to teach you how to do a little bit of exponents.  some key notes are 1) b^x * b^y= b^x+y 2)b^x / b^y= b^x-y 3) #^0= 1 4) (ab)^x= a^x * b^y 5) (a/b)^x= a^x/ b^y
these are just some common notes that will be used in the examples below.


EX:
1) (-4)^-2
-1/4 *-1/4
= 1/16
2) -4^-2
=-1/16
3)(5*2)^-3
5^-3 * 2^-3
=1/125 * 1/8= 1/1000

EXPONENTS!

This week we learned about exponents. An exponent is A quantity representing the power to which a given number or expression is to be raised, usually expressed as a raised symbol beside the number or expression. The first thing i'm going to do is give you some laws about exponents. Below i made a chart of some of the basic laws involving exponents with examples next to each law.

Law	Example
x1 = x	61 = 6
x0 = 1	70 = 1
x-1 = 1/x	4-1 = 1/4
xmxn = xm+n	x2x3 = x2+3 = x5
xm/xn = xm-n	x6/x2 = x6-2 = x4
(xm)n = xmn	(x2)3 = x2×3 = x6
(xy)n = xnyn	(xy)3 = x3y3
(x/y)n = xn/yn	(x/y)2 = x2 / y2
x-n = 1/xn	x-3 = 1/x3
And the law about Fractional Exponents:

Here's a few more examples of exponents.

Example 1: (5+2)^2

7^2 which is 49

Example 2. 5^-2

1/5^2 which is 1/25

Example 3. 2^2/2^3

2^-1 which is 1/2

Example 4. (3^2)^3

3^6 which is 729

that's about it on exponents. Just remember the laws and you're good to go!

--Carleyyy

13-5

Okay, so this week I am going to teach you how to work infinite sum problems. They are very easy to learn how to do and are mostly algebra. But, first I am going to tell you a few things you are going to need know.

Notes:

• You can't find an infinite sum of an arithmetic series.
• Only geometric series where lrl < 1 have an infinite sum.
• The formula for these types of problems are S=t1/1-r
• When finding the infinite geometric converges, set lrl < 1 and solve for x.
• The last thing you need to know is, to write a repeating decimal as a fraction you would do what's repeating/place - 1

Okay, so now that you know all of that I am going to work a few examples for you. This should help you better understand what is going on.

Example 1: Write this repeating decimal as a fraction .23232323

• The number that is repeating is 23
• So you would put 23/100 -1
• You would put 100 - 1 becaue the 3 is in the 100ths place.
• So you would then get 23/99.
• And your answer is 23/99.

Example 2: Find the sum of the infinite geometric series.. 24-12+6-3+...

• You are going to use the formula I gave you above which is S=t1/1-r.
• S is what you are trying to find which is the sum.
• t1 is the first term given to you, which is 24
• r is the number that is being mulitplied to get the next number, which is (-1/2).
• So then you are going to plug into the formula.
• S= 24/1-(-1/2)
• Your answer is going to be S=16

Well that's it for this week. Byeeee

--Halie! :)

13-5

13-5 Sums of Infinite Series

-In this lesson, we learned how to find the sum of an infinite geometric series. Notice, I didn't include arithmetic; You can't find an infinite sum of an arithmetic series.

-Only geometric series where lrl < 1 have an infinite sum. Formula: S = t1/1-r

-In order to find were an infinite geometric converges, set lrl < 1, and solve for x.

-To write a repeating decimal as a fraction: what's repeating/place - 1

Example 1: Find the sum of the infinite geometric series 24 - 12 + 6 -3 +...

• S = t1/1 - r
• S = 24/1 - (-1/2)
• S = 24/1 + 1/2
• S = 16

Example 2: For the infinite geometric series, find the interval of convergence.

• 1 + x^2 + x^4 + x^6 +...
• lx^2l < 1
• -1 < x^2 < 1 (square root all three terms to get your answer)
• -1 < x < 1

Example 3: Express the given repeating decimal as a rational number; o.7777...

• 7/10-1 (the repeating decimal ends in the 10th's place)
• 7/9

Sequences

This is also a pretty easy concept to pick up on.

A sequence is a list of numbers which form a pattern. We are going to look at two types of sequences: Arithmetic and Geometric.
An Arithmetic Sequence is a sequence that is characterized by adding the same number each time.
The formula for arithmetic sequences is
tn=t1 + (n-1) x d
d is what is added or the difference.
tn is __ terms

A Geometric Sequence is a sequence that is characterized by mulitplying the same number each time.
The formula for geometric is
tn=t1 x r^(n-1)
r is whatever is being mulitplied
n is the term number
tn is __ terms


Ex 1: Find the tn formula for the following sequence:
2,5,8,11
It is arithmetic since 3 is added each time. So we plug it into the tn formula.
Tn=2+(n-1)3
Tn=3n-1

Ex 2: Find the tn formula for the following sequence:
2,6,18,54

It is geometric since everything is multiplied by 3.
Tn=2(3)^n-1


-Sameer

13-3

Well as we all know this week we started chapter thirteen. Anyway section three of chapter thirteen is what I will be reviewing in this blog.
-To find the sum of the ____ terms of an arithmetic series:
Sn=n (t1 + tn)/1-r
-To find the sum of the ____ terms of a geometric series.
Sn= t1 (1-rn)/1-r
-A series is a list of numbers that are added together.
2+4+6+8
EXAMPLE TIME BOYS AND GIRLS
Find the sum of the first 8 terms of the arithmetic series.
11+14+17+20+…
First of all, set up the equation
S8= 8 (11 + )/2
To find tn, use your knowledge of the previous sections. T8= 11 + (n-1) (3)
T8= 11+3n-3 = 11 + 24-3= 25-3
T8=22
Plug this number into your equation next.
S8= 8 (11 + 22)/2= 132
That is about all there is to it. Do not forget your assignments that are due tomorrow! Have a fantastic week!

--SARAH (:

13-4 infinite limits

today im teaching you about infinite limits. although this sounds hard it is really easy. it is easier then most stuff we have done this year. if you have a fraction with 2 polynomials divided use these rules...  1) (degree) top = (degree) bottom  lim x --> (infinity) = lead coeff. / leading coeff.  2) (degree) top > (degree) bottom  lim x --> (inifity) = +/- (infinity)  3) (degree) top < (degree) bottom   lim x --> = 0.   if it doesn't follow rules... 1) plug into y=   2) 2nd table  3) plug in 10|100|1000|10000 until you see pattern. 4) E +ve # = + or - inf  E -ve # = 0.  if it is geometric 1) |r| < 1 then = 0 if |r| > 0 = (infinity)


EX: lim n --> (infinty)   n+5/ n
 look at all the rules
top = bottom
1/1 = 1

13-1 Arithmetic and Geometric Sequences

This weekend I am going to teach you all about arithmetic and geometric sequences. These are patterns that you have to do different things with. Sometimes they will ask you for a certain term number and other times they will ask you for a formula. There are some things that you need to know about arithmetic and geometric sequences before I show you all an example.

Notes:

• Arithmetic Sequences
o They are generated when the same number is added every time
o The formula for this type of sequence is tn = t1 + (n-1) * d
o d= what is being added, or the difference
o tn= ___ term
o t1= 1st term
o n= term number

• Geometric Sequences
o They are generated when the same number is multiplied every time
o The formula for this type of sequence is tn = t1 + r^(n-1)
o You cannot divide, so you have to multiply by fractions
o r = what is being multiplied

Example: tn= 2n + 5; find the first four terms and state whether the sequence is arithmetic or geometric
• 2 (1) + 5 = 7
• 2 (2) + 5 = 9
• 2 (3) + 5 = 11
• 2 (4) + 5 = 13
• 7, 9, 11, 13,…..
• This sequence is arithmetic; +2

-Braxton

Sequences!

So this week we are gonna learn about two different types of sequences. A sequence is a list of numbers with a pattern. Today we are going to learn about Arithmetic and Geometric sequences.

An Arithmetic Sequence is a sequence that is generated by adding the same number ech time.
The formula for arithmetic is
tn=t1 + (n-1) x d
d is what is being added or the difference.
tn is __ terms

A Geometric Sequence is a sequence that is generated by mulitplying the same number each time.
The formula for geometric is
tn=t1 x r^(n-1)
r is what is being mulitplied
n is the term number
tn is again __ terms

Now that you know some information on this, we're going to work some problems.

Example 1. Find the tn formula for the following sequence:
1,4,7,10
This is arithmetic because you add 3 each time. So we plug it into the tn formula.
Tn=1+(n-1)3
Tn=3n-2

Example 2: Find the tn formula for the following sequence:
2,4,8,16
This is geometric because you multiple everything by 2.
Tn=2(2)^n-1

I hope you learned about sequences. See you next week!

--Carley!

arithmic and geometric sequences

soo this week we learned about arithmic and geometric sequences. they are easy once you know the formulas and what the different terms aree.

Here are some things for arithmic

• Arithmetic Sequence is a sequence that is generated by adding the same number ech time.
• The formula for arithmetic is tn=t1 + (n-1) x d
• d= what is being added or the difference.
• tn= blank terms

Here are the things for geometric

• Geometric Sequence is a sequence that is generated by mulitplying the same number each time.
• The formula for geometric is tn=t1 x r^(n-1)
• r= what is being mulitplied
• n=term number
• tn=blank terms

Here is an example of an arithmetic sequence

tn=2n+2
2(1)+2=4
2(2)+2=6
2(3)+2=8
2(4)+2=10

Here is an example of a geometric sequence

tn=2x3^n

2x3^1=6
2x3^2=12
2x3^3=24
2x3^4=48

bam! all done!

13-1

In 13-1 you will be doing sequences. A sequence is a list of numbers. there are two types of sequences: arithmetic and geometric. An arithmetic sequence is a sequence that is generated by adding the same number each time. A geometric sequence is a sequence that is generated by multiplying the same number each time. The formulas for the two sequences are as follows:

geometric sequence

tn= t1 + (n-1)(d)

t1 is the first term. N is the term number. D is the difference, what is being added. Tn is is nth term.

arithmetic sequence

tn= t1 * r^(n-1)

T1 is the first term. R is the term that is being multiplied. N is the term number. Tn is the nth number.

Example 1:

Find a formula for the nth term 3,5,7.

First, you have to figure out if the sequence is arithmetic or geometric. To get from term to term you have to add two, making the sequence arithmetic.

Once you've figured out the type of sequence, you have to use the arithmetic formula to find the nth term.

tn= 3 + (n-1)(2)

tn= 3 + 2n - 2

tn= 2n+1

The formula for the nth term is 2n+1.

Example 2:

Find a formula for the nth term 4,8,16,32

To get from term to term you have to multiply by 2, making it a geometric sequence.

tn= 4 * 2^(n-1)

tn= 4 * 2^n * 2^-1

tn= 4 * 2^n / 2

tn = 2 * 2^n

The formula for the nth term is 2 * 2^n.

13-1

Things you should know:

• A sequence that is generated by adding the same number each time is called the arithmetic sequence.
• Formula: tn = t1 + (n-1) x d
• d = what is being adding/difference
• tn = blank terms
• t1 = first term
• n = term number
• A sequence that is generated by multiplying the same number each time is called the geometric sequence.
• Formula: tn = t1 x r^(n-1)
• r = what is being multiplied
  
• n = term number
• tn = blank terms
• t1 = first term
• A sequence is a list of numbers.

Example 1: tn = 6n + 20. What is the sequence? Is it geometric, arithmetic, or neither?

• First thing you need to know is you plug in the number of terms you want to find out for n.
• You are going to find out the first four terms.
• T1 = 6(1) + 20 = 26
• T2 = 6(2) + 20 = 32
• T3 = 6(3) + 20 = 38
• T4 = 6(4) + 20 = 44
• Now we have to find out what is the sequence.
• Each time you are adding 6 to get to the next number so that means it is an arithmetic sequence.

Example 2: tn = 5 x 10^n-1. What is the sequence? Is it geometric, arithmetic, or neither?

• Just like in example one, you have to plug in the number of terms you want to find our for n.
• You are going to find the first four terms again.
• T1 = 5 x 10^1-1 = 5 x 10^0 = 5
• T2 = 5 x 10^2-1 = 5 x 10^1 = 50
• T3 = 5 x 10^3-1 = 5 x 10^2 = 500
• T4 = 5 x 10^4-1 = 5 x 10^3 = 5000
• Now you have to find out the sequence.
• Each time you are multiplying 10 to get to the next number so that means it is an geometric sequence.

- Amber :)

13-1

13-1 Arithmetic and Geometric Sequences

In this lesson, we learned how to identify the two different types of sequences, arithmetic and geometric, and how to find a formula for its nth term.

• arithmetic sequence: a sequence that is generated by adding the same number each time

-formula: tn= t1 + (n-1)d; t1= 1st term, n= term #, d= difference(what's being added), and tn= _term

• geometric sequence: a sequence that's generated by multiplying the same number each time *to divide, we use fractions (/2 = x 1/2)

-formula: tn= t1 x r^(n-1); t1= 1st term, r= what's being multiplied, n= term #, and

tn= _term

Example 1: Is the following an arithmetic or geometric? 3, 5, 7, ...; Find the formula for the nth term.

-It's arithmetic because you're just adding two to each number.

-tn= t1 + (n-1)2

= 3 + (n-1)2

= 3 + 2n - 2

= 2n +1

Example 2: For tn= 5n + 2, find the first 4 terms and state if it's arithmetic or geometric.

1) 5(1) + 2 = 7

2) 5(2) + 2 = 12

3) 5(3) + 2 = 17

4) 5(4) + 2 = 22

-It's arithmetic because you can see that 5 is being added to each number.

13-1 Arithmetic and Geometric Sequences

Okayyy, so this week I am going to teach you how to work arithmetic and geometric sequence problems. It might seem complicating at first, but it is actually very easy and quick to learn. First, i have to give you some basic notes and formulas that you are going to need.

Notes:

• Arithmetic Sequence is a sequence that is generated by adding the same number ech time.
• The formula for arithmetic is tn=t1 + (n-1) x d
• d= what is being added or the difference.
• tn= blank terms

• Geometric Sequence is a sequence that is generated by mulitplying the same number each time.
• The formula for geometric is tn=t1 x r^(n-1)
• r= what is being mulitplied
• n=term number
• tn=blank terms

A sequence is a list of numbers.

Okay, so now that you know all what you need to, I am going to work a few examples for you.

Example 1: tn=2n + 3. Figure out the sequence and if it is geometric, arithmetic, or neither.

• You would plug in the number of terms for n.
• 2(1) + 3 = 5
• 2(2) + 3 = 7
• 2(3) + 3 = 9
• 2(4) + 3 = 11
• From that you can tell that the sequence is arithmetic because it is adding 2 to get to the next number.

Example 2: tn=3 x 2^n. Figure out the sequence and if it is geometric, arithmetic, or neither.

• You would do the same thing as in example one.
• 3 x 2^(1)= 6
• 3 x 2^(2)= 12
• 3 x 2^(3)= 24
• 3 x 2^(4)= 48
• From that you can tell that the sequence is geometric because it is mulitiplying 2 to get the next number.

So that is how you work those types of problem. Byeee.

--Halie !