Answers by visitors named: Francesca
Yes this is a multiple question. I think the answer is choice 1 but im not sure.
- Francesca
By the way, my friend Jonathon, thinks it's 2. What do you think?
I think it is 1. But i'm not sure if choice 2 refers to the Byzantine Empire, which in that case, I think it would be 2.
I heard of them but i'm not exactly sure about them.
Thank you for responding. Hmm... IDK. . .I'll have to ask if that's a typo on the other end. Hey do you mind seeing if this is correct, and helping with the second part?
Consider the geometric sequence that begins -3072 and common ratio –1/2.
Find the 13th and 20th terms of this sequence.
a₁₃ = -3072(-1/2)¹² = -3/4
a₂₀ = -3072(-1/2)¹⁹ = -0.005859375
Is this right?
b. Find the sum of the first nine terms.
I'm not sure what to do here. . .
Thank you again for your help!
OK thank you!
Thank you!
OK Thank you!
ok thank you!
Well, I am suppose to find the composition of functions from a figure. If you don't mind I uploaded a photo of it on photobucket. Can you take a look, and offer any suggestions for the first problem, so that I can get an idea? Since this forum will not let me post a link I can give directions on how to find the photo. First, go to search bar and type: flutegirl516. Then, this message will appear: Are you looking for the Photobucket user flutegirl516? Click on this. Then you will see the photo album. There is only one picture.
Also, click "View as slide show" to make it larger. Tell me if you can view it okay. Thanks for any helpful replies :)
I read the the link you provided previous to posting this discussion, but I was still confused.
I'm using the search bar at the very top right-hand corner, then I enter: flutegirl516. It will say no matches found, but it will say: Are you looking for the Photobucket user flutegirl516?
You can also try using the search bar drop down menu and clicking on "Users," and then enter flutegirl516. The album should pop up. Try it, hopefully you will be able to view it.
ooOo I think I get it now. . .
Ok so
g ° f = {(1, 4), (2, 2), (3, 2), (4,3)}
There are two 1s in the range of f(x) though (1,2) and (3,1). . .Does that mean anything?
Is this correct:
Does g ° h = {(1, 2), (2?, 2?), (3, 2), (4, 1)}? For this one there was no 2 in the range of h.
Also, for h^2 = h ° h, what am I suppose to square? Domain (x) or Range (y).
"There are two 1s in the range of f(x) though (1,2) **I meant to say (2,1)** and (3,1). . .Does that mean anything?" Anyway, scratch this statement of the previous post. . .
Here are the functions from the figures:
• f(x) = {(1, 2), (2,1), (3,1), (4, 4)}
• g(x) = {(1, 2), (2,4), (3, 1), (4, 3)}
• h(x) = {(1, 1), (2, 3), (3, 1), (4, 3)}
So is this correct?
g ° f = {(1, 4), (2, 2), (3, 2), (4,3)}
There is something not connecting in my thought process. I'm still confused with g ° h does it = {(1,2), (3,1), (1,2), (4,3)}? I feel way off. . .I am doing something terribly wrong. IDK
Is this correct?
g ° h ={(1,2), (3,1), (1,2), (4,3)}
Lol. . .But really Thank You! You are really helping me to understand. I know you are probably annoyed by my silly questions, but I am really starting to get a better understanding.
Ok so would h ° h = {(1,1), (2,1), (3,1), (4,1)}? But the h² is throwing me off.
They appear a little strange. . .how so? The h² does not effect the answer at all?
OK so last one f ° g ° h. . .I will attempt now, and check back and see if I am on the right track. . .
Oh it looks strange b/c they all end in zero. Just noticed. . .
Is this correct?
f ° g ° h = {(1,1), (2,2), (3,1), (4,2)
Hey, I know I said the previous would be the last one but can you check this one too. . .
Let f: A→B be a function from A to B. f = {(w, 1), (x, 2), (y, 3), (z, 2)}. Find f^-1.
Answer: f^-1 = {(1, w), (2, x), (3, y), (2, z)}
How about this?
Let f: A→B be a function from A to B. f = {(w, 1), (x, 2), (y, 3), (z, 2)}. Find f^-1.
Answer: f^-1 = {(1, w), (2, x), (3, y), (2, z)}
So would this one not have an inverse?
Oh okay we posted at the same time. . .You were such a big help! I really understand this stuff a lot better. Thank you a thousands times!
Also thank you for the time you took out to help me :)
So, if A = (2,4,6) and B = (21,42,52)
g : B -> A, wouldn't g = {(21,2), (42,4),(52,6)}? Is g ° f defined because it is one-to-one?
Figured this one out already had to use the equation:
x = v₀x√(2h/g)
Can you give an example using numbers?
So, if A = (1, 2, 3) and B = (4, 5, 6)
f: A -> B => {(1, 4), (2, 5), (3, 6)}
g: B ->A => {(4,1), (5,2), (3,6)}
g ° f = {(4,4), (5,5), (6,6), so g ° f = B right? And the domain and range are equal. But I'm still sure how it is defined?
When finding the composite of a function you are suppose to match the range to the domain. If you go to photobucket and type in flutegirl516 in the search bar there is a picture that explains it better. It's the photo with the colorful lines. . .
I received that picture from my teacher. . .that's what she says is correct. That's why I am so confused. . .I didn't just make that up. . .So I really don't know what to follow, but I guess if I want to get a good grade I better just go along with the teacher. The way you explain it makes sense though!
Okay thank you
Thank you for replying. So if this question makes any sense: How do you know I must start with G first? It makes sense because it actually works when I create samples, but it seems that I am working backwards, why is that? But I think I am kind of following. . .
So, how would F ° h ° G be defined? It seems the domain is A and the range is C, right? But why does it exist? Does it exist because it can be proven using sample problems or it it because of the figure and the direction of the arrows? IDK. That's what I am having trouble understanding.
Okay thank you for the clarification :)
So, this would read as :
_ __
2 = 18 (mod 8)
Meaning this would be considered true, right?Because 18/8 = 2, and then 8 · 2 = 16, making the remainder 2.
This might sound silly, but what does the symbol at the top of the number mean?
_
2
2 ∈ 18 (mod 8)
It is true though, right?
Hmmm...I kind of get what you are saying, but why is 18 not a set that does not include 2?
Here is an example in the book that is true:
_
55 ∈ 7 (mod 3)
_
7 (the line goes over 7 in the above)
Why would this be considered true?
Could you tell me if I am correct in thinking:
• With respect to congruence mod 29, 17 ∩ 423 = ∅ (True)
• Let a, b, and n be integers with n > 1. Then a ≡ b (mod n) ⇔ a = b (False)
•If ac ≡ bc(mod n), and gcd(c, n) = 1, then a ≡ b(mod m) (True)
I think I found something about the overbar
_
a <--- equivalence class of a
_
b <---equivalence class of b
OK. . .If you don't mind how about the these two too:
• With respect to congruence mod 29, 17 ∩ 423 = ∅ (True)
•If ac ≡ bc(mod n), and gcd(c, n) = 1, then a ≡ b(mod m) (True)
Yea I was thinking that too, that it is the same thing,but I will double check with the teacher.
So,With respect to congruence mod 29, 17 ∩ 423 = ∅ would be considered false, right?
I am too guilty of double posting disregard above 11:51
Last thing I want to ask. . .
5x ≡ 5(mod 25)
Is there an easier way to derive to the answer. Because I believed I learned the long version.
This is what I know:
The possible values are 0, 1, 2, 3 . . .24
5(0) - 5 = -5 not divisible by 25
5(1) - 5 = 0 not divisible by 25
5(2) - 5 = 5 not divisible by 25
5(3) - 5 = 10 not divisible by 25
So, I would this until I calculated a number that is divisible by 25, but there has to be an easier way, right? I think I saw something online where they found gcd. . .IDK though. If you're up to it can offer an easier way to decipher what x will equal?
So, x = 1? Or can it be multiple answers? But what if I have a big equation like:
4x ≡ 320(mod n), n = 592
So, once I reduce it to this:x≡80 (mod 148). Then I have to do the trial and error process? Idk, I get how you got 80, but how did get 228, 376, 524. . .I think I see a pattern though each are incremented by 148. Hmm. . .I have another example that I would like to share, but I am going to work it out first, then post back in a few minutes to see if I am on the right track.
So, I worked out a couple. . .
1.) 4x ≡ 2(mod 6)
x = 0, 4(0) - 2 = -2 is not divisible by 6
x = 1, 4(1) - 2 = 2 is not divisible by 6
x = 2, 4(2) - 2 = 6 is divisible by 6
x = 3, 4(3) - 2 = 10 is not divisible by 6
x = 4, 4(4) - 2 = 14 is not divisible by 6
x = 5, 4(5) - 2 = 18 is divisible by 6
x = {2, 5}
2.) 4x ≡ 4(mod 6)
x = 0, 4(0) - 4 = -4 is not divisible by 6
x = 1, 4(1) - 4 = 0 is not divisible by 6
x = 2, 4(2) - 4 = 2 is not divisible by 6
x = 3, 4(3) - 4 = 8 is not divisible by 6
x = 4, 4(4) - 4 = 12 is divisible by 6
x = 5, 4(5) - 4= 16 is not divisible by 6
x = {4}
3.)5x ≡ 5(mod 5)
x≣ 1 (mod 1)
x = {1}
Are these correct? If not, what am I doing wrong?
7x ≡ 3(mod n), n = 11
No x exist, right? But why?
Also, back to the very first one:
5x ≣ 5(mod 25)
x ≣1 (mod 5)
x = {1, 6, 11, 16, 21} <---Is this the whole answer?
OK you gave me a lot to think about. . .Thank you so much for your help. Until next time (which may be tomorrow). Thanks again :)
Thank you so much for your response! But I have completed that particular question. However, can you please help with this one? I am confused. . .
Use mathematical induction to establish the following formula.
n
Σ i² / [(2i-1)(2i+1)] = n(n+1) / 2(2n+1)
i=1
Thanks for any helpful replies :)
Any suggestions?
Ok thank you for your helpful response! I have a couple of questions though. . .
Is the 15th line suppose to be '(k+1)(k+2)/(22k+3)'?
Also, the 16th line = RS, which is what exactly?
Yea that's what I thought. . .Hey if you don't mind helping me further I have been working on this problem for a while and I am a bit stuck. IDK where to go from here or if I am doing it correctly:
Use mathematical induction to prove the truth of each of the following assertions for all n ≥1.
5^2n – 2^5n is divisible by 7
If n = 1, then 5^2(1) - 2^5(1) = -7, which is divisible by 7. For the inductive case, assume k ≥ 1, and the result is true for n = k; that is 7 | (5^2k + 2^5k). Use the assumption to prove n = k + 1, in other words, 5^(2(k + 1)) - 2^(5(k + 1)) is divisible by 7. Now,
5^(2(k + 1)) - 2^(5(k + 1))
= 5^(2k + 2) - 2(5k + 5)
= 5^(2k) · 5^2 - 2^(5k) · 2^5
= 25 · 5^(2k) - 32 · 2^(5k)
= IDK what to do from here. . .
Any suggestions? Thank you again!
Any suggestions?
Thank you for responding. Yes everything is typed correctly. I want to find what is wrong with proof.
No, the question verbatim is "What is wrong with this proof?"
Thank you! So, going back to your counterexample in post 9:52:
x=4, y=6,
n=max(x,y)=6
Why does it =6? Sorry if this seems like a silly question. . .
Oh okay. . .I get it. . .Thank you so much for your help :)
Here is the solution: The mistake is in applying the inductive hypothesis to look at max(x −1, y −1) . Notice that we are doing induction on n not on x or y. Even though x and y are positive integers, x −1 and y −1 need not be (one or both could be 0). In fact, that is what happens if we let x = 1 and y = 2 when k = 1.
I have to admit I do not quite understand this explanation. . .
Ok thank for the responses, but there seems to be a contradiction between the two. Wouldn't f(1) = 1 + 2, which equals 3?
The f(n+1) is throwing me off what does that mean?
OK example: f(n+1) = 3f(n)
f(1) = 3
f(2) = 6
f(3) = 9
Right?
Oops. . .Sorry disregard previous post. . .
f(n) is defined recursively by f(0) = 1 and for n = 0, 1, 2, . . .
Find f(1), f(2), f(3)
Yes, they both follow the same recursive definition. I was just trying the second part on my own to see if I understand. Sorry about the misunderstanding. . .
So, this is how far I got. . .I getting weird numbers. . .
-3072(1 - (-1/2)⁹)
------------------- =
1 - (-1/2)
REVISED QUESTION: Why use mathematical induction to prove the sum of a sequence is valid?
Sorry I still don't get it. Can someone please explain?
If you don't mind can you help with this problem?
Solve the recurrence relation an+1 = 7an – 10an - 1, n ≥ 2, given a₁ = 10, a₂ = 29.
The characteristic polynomial is x^2 - 7 + 10 with characteristic roots 2 and 5. Once again I get confused when I get to this part. We obtain a_n = C_1(2)^n + C_2(5)^n.
a_1 = 3C_1 + 3C_2 = 10
a_2 = I don't know what to put here = 29
Oops posted twice. . .Sorry
Thank you so much for your help! I think I am getting the hang of it better. Can you please check:
Solve the recurrence relation a_n = -5a_n - 1 + 6a_n - 2, n ≥ 2, given a₀ = 5, a₁ = 19.
characteristic polynomial is x^2 + 5x - 6 it has the distinct root 2 and 3.
a_n = C_1(2)^n + C_2(3)^n
a0 = C_1 + C_2 = 5 (1)
a1 = 2C_1 + 3C_2 = 19 (2)
So to find C2 I eliminated it by (2) - 2(1)
(2C1 + 3C2) = 19
- (2C1 + 2C2) = 10
___________________
C2 = 9
Plug this into (1) and this is where I get confused because I get a C1 = -4, is it suppose to be negative? Am I doing a step wrong?
a_n = . . . (?)
Oh I feel dumb. . .
Ok so now
a_n = C_1(1)^n + C_2(-6)^n
a0 = C_1 + C_2 = 5 (1)
a1 = C_1 - 6C_2 = 19 (2)
So to find C1 I eliminated it by 6(1) + (2) <--is this allowed?
(6C1 + 6C2) = 30
+ (C1 - 6C2) = 19
___________________
7C1 = 49
C1 = 7
Plug this into (1) and this is where I get confused because I get a C1 = -2, is it suppose to be negative? Am I doing a step wrong?
a_n = . . . (?)
Idk what happened at 7:24 I think my computer had a glitch or something and it reposted.
This one is throwing me for a loop:
Solve the recurrence relation a_n = 2a_n - 1 – a_n -2, n ≥ 2, given a₀ = 40, a₁ = 37.
Characteristic polynomial: x^2 - 2 + 1
How do I find the roots?
So the characteristic roots are 1?
How about this:
Solve the recurrence relation a_n+1 = -8a_n – 16a_n - 1, n ≥ 1, given a₀ = 5, a₁ = 17.
Characteristic polynomial is: x^2 + 8x + 16 with distinct roots -4.
Since the roots are equal a0 = 5 = C1(-4)^0 + C2(0)(-4)^0 making C1 = 5, right? But when I apply a1 = 17 = C1(4) + C2(1)(-4), it does not work for me. . .I always seem to get confused
So, going back to the previous 9:05. The solution is a_n = 40(1)^n - 3(1)^n, is this correct or way off?
Well a_1 = 37 not 31
Hey thanks a lot for help!
Okay I continued the first problem:
|A ∩ B| = [2000/6] = 333
|B ∩ C| = [2000/15] = 133
|C ∩ D| = [2000/35] = 57
|A ∩ D| = [2000/14] = 142
|A ∩ B ∩ C ∩ D| = [2000/210] = 9
1000 + 666 + 400 + 285 - 333 - 133 - 57 - 142 + 9 = 1695 <--Answer
Any suggestions?
To be honest I haven't started yet, but your method sounds like a step in the right direction. . .I'll play around with it for a little and see what I get. . .If you figure out anything post. Hopefully someone who knows something will post cuz I'm lost
Oh okay I think I am following. . .
So, how many seven-letter palindromes contain at most three different letters one of which is S?
We would start out with 26^3, but I don't understand how to make sure S will be included as one of the different letters. Any suggestions? Thank you.
Oh ok so there are 13800 that contain at most three different letters one of which is S.
Can you at all help with this?
Multiple personality disorder (MPD) is a condition in which different personalities exist within one person and at various times control that person’s behavior. In a recent survey of people with MPD, it was reported that “98% had been emotionally abused, 89% had been physically abused, and most had experienced both types of abuse.” Make this statement more precise.
Ok now I am confused again with 5:08, it seems the same as the first subpart of the problem. They can't be the same answer, for this one: How many seven-letter palindromes contain at most three different letters one of which is S?
And for the MPD problem that was really the whole problem. It seems like I have nothing to work with though. . .I'm not sure where to go from here
You can consider
n: number of people surveyed.
E: the set of those who had been emotionally abused.
P: the set of those who had been physically abused.
Basically for the MPD problem I have to make it more precise.
THANK YOU! :)
588 is the correct answer? But, I don't understand how to get to that number. How did you calculate that? Sorry, that may be a lot to type out
588 is the correct answer!**
Yea I tried doing a google search, but nothing good explaining where to start. Thnx for the reply though
Sorry it's computing in security, so I guess that would fall under computers. . .
Thanks for your help!
Also, I am kind of confused about finding the value of x.
Example: 9x ≡ 1 mod 10
How do I solve this?
Any help?
Thank you guys!
Thank you!
1.) Find the producers' surplus if the supply function is: S(q) = q^7/2+3q^5/2 + 54. Assume the supply and demand are in equilibrium at q= 25.
2.) S(q) = q^2 + 12q and D(q) = 900 - 18q - q^2
The point at which the supply and demand are equilibrium is (15, 405).
a.) Find the consumers' surplus
b.) Find the producers' surplus
Thank you!
Also, how would you calculate the IRR? Thank you for any help!
The given are (a) the speed of sound, which is 395 m/s, (b) the velocity of the moving source, which is 12 m/s, and (c) the frequency of the sound both cars are emitting, which is 395 Hz.
Let's assume that f{o} is the frequency the observer can hear and f{s} is the frequency the source emits, Vs is the velocity of the moving source, and v is the speed of sound.
The equation for a source moving towards an observer is f{o} = f{s} x (1- (Vs÷v))
f{o} = 395 Hz x (1 - (12 m/s ÷ 343 m/s))
f{o} = 381.180758 Hz
To compute for the beat frequency, let's assume that f{beat} is the beat frequency:
f{beat} = f{a} - f{b}; where f{a} is the larger of the two.
f{beat} = f{a} - f{b}
f{beat} = 395 Hz - 381.180758 Hz
Thank you!