5

Use induction to prove 2(1+3+3 2 +3 2 +···+3 n−1 ) = 3 n −1...

Question

Use induction to prove 2(1+3+3 2 +3 2 +···+3 n−1 ) = 3 n −1

Use induction to prove 2(1+3+3 2 +3 2 +···+3 n−1 ) = 3 n −1



Answers

Prove the following by using the principle of mathematical induction for all $n \in \mathbf{N}$. $$ 1+3+3^{2}+\ldots+3^{n-1}=\frac{\left(3^{n}-1\right)}{2} $$

Okay. In this problem I want to prove that three is a factor of in Q plus three. N squared plus two in. For all in greater than or equal to one. So let's think about what that means for a minute. That what kind of saying is no matter what number I put in here. Eight or 15 or 10 or whatever. This number here in Q plus three N squared plus two and will be a multiple of three. Alright, so here we go. We're going to prove it by induction. So proof by induction. First step. Show that it's true. For N equals one. Show it is true. Yeah. For any close one. Mhm Okay. For an equals one in cube plus three, N squared plus two in equals one cubed plus three times one squared plus two times one. That's one plus three plus two, which is six. Three is a factor of six. Therefore true. Uh huh. I'm not gonna put that. I'm going to say I'm done right there shows true. For n equals 13 is a factor of six. So when I plugged in and equals one, I get a number which is six and three goes into it. So now I'm done with that part Step two. Yeah, I assume it is true for an equals K. That is I assume three is a factor of K cubed plus three K squared plus two K. Yeah. Okay. So before I leave that sense, I've got to say what what that means. I gotta make that into some sort of math thing. Well three is a factor six because there exist a number two. Such that when you multiply three by it, you get 63 times two is six. So if three is a factor of this thing, KQ plus three K squared plus two K. Then there's some number, let's call it M. And three times this M will give you this cake. You plus three K squared plus two K. King. So there is. And into your whole number. Okay. Because we're only worrying about positive things. They're not. And hole number. There is. Yeah. Whole number. Mm. Such that K cubed plus three K squared plus two K equals three M. There's some number M. That you multiply by three and you get that thing. Okay now show it is true for N equals K plus one. That is show three is a factor of K plus one, cube plus three K plus one squared plus two times K plus one. Okay slap means show there is a whole number can we can't call it? Um There's a whole number something. Let's call it. Uh L. Such that. That's okay. Plus one cubed plus three times K plus one squared. Mhm. Plus two times K plus one equals three. L Alright here we go. So now here comes the mouth part. I'm going to start with the left hand side of this just looks just like I always do. And then I'm going to try and get the left hand side of this one in there. And once I get that in there, I'm gonna replace it with three a.m. And then see what we can get. All right? So here we go. Yeah, K plus one cubed plus three times K plus one squared plus two times K plus one equals all right. So now I got to decide what to do next. Do I want to factor out A K. Plus one? Or do I want to multiply all these things out? And my guess is I need to multiply all these things out so I can get this right here in there because that will be the first term of each thing after I get it multiplied out. All right. So there you go. I'm doing K plus one cubed is K cubed plus three K squared plus three K plus one plus. Now I'm doing three times K plus one squared. So that will be three times K squared plus two K plus one plus two K plus two. So Kay cubed plus three K squared plus three K plus one plus. Notice. I have not combining anything yet. Plus two K plus two. All right. So now I see here I have a K cube A three K squared plus two K. And I know that from a pier that stuff is equal to three a.m. So I'm gonna rewrite it so you can see me put it in there. So this is equal to K cubed plus three K squared plus two K plus. Now, what do I have left Once I take that out? I have a three K square I have three K. And a six K. Which is nine K. And I have a one and a three and a two. So plus six. All right. And so that is equal to three M plus three K squared plus nine K plus six. Okay. Now look at the numbers in front of each term here, I have a three A three, A nine and a six. I know that three divides three M. Three divides three K squared, three divides nine K. Three divide six. So three divides this whole thing. So here's how I'm going to show you that when you factor out the three? Yeah. Yeah. All right. And remember, all I said was I'm trying to show that that thing is equal to three times some L right here. So I'm gonna say let that stuff bl so that equals three ill. Mhm. For L equals M plus k squared plus three K plus six. Therefore, three is a factor yeah, of K plus one. Cute plus three K plus one squared. Mhm Plus two K plus one. And so right, three is a factor of k cubed plus three K squared plus two came

So prove this. Using induction, we first have to look at the base case and equals one. So we haven't left inside. Our first term is to That's gonna be equal to end just one times. Three in so three times one plus one all divided by two. So we solved this. It's a pretty easy the coefficient of one and then we have three plus one is four divided by two. So electric, too. We can see pretty easily that the base case works for this induction. Now we have to look at the inductive case. So given that is equal, okay. Me? No, the two plus five plus a whole bunch of other numbers with the same pattern gonna be equal three and minus one, and and in this case is gonna be okay. And that's equal to and okay, times three K minus one. Invited by two. So we go about solving this by adding on the next term. So mathematically, this is shown by adding on she's representative Red three times K plus phone minus form, term to the end of our left hand side are some of all these terms No notice. This is just the equivalent for induction step with and equal. The cape looks one K plus one instead of and is equal. Okay, so we continue to get that equals three and three K plus one, all divided right to. And then we have a plus three three K plus one minus one in that second threes. Mistake that she will be there. No, Keep going. What I'm gonna do is I'm gonna put an extra two in front of all of our terms that we just added. So in other words, okay. Three K plus one, all divided by two plus. So we have to times three. Okay. Plus three minus one, all divided by two. That's just to combine the like terms so we can get one fraction said to. So we get we factoring. Okay. Three K squared plus K plus six k from the right hand side plus six minus two, divided by two. And we're left with Greek. A square plus seven K plus four divided by two. So I'm going to do next. Is I'm just gonna try a little trick. I'm going with you. Split up that middle term seven k. Thank you. Three k plus four K and that will be known in a second. So we continue going now we can factor out of three K from the top. You are the left hands two terms specifically these two and gets three k times K plus one. Then we have plus who fact out of four. On the right hand side, we get K plus one. Now we have a distributive property with K plus one as the term that were factoring out of these two gets okay, plus one times three K plus four provided by two. And then way dio K plus one. And now we take three digits out of our four. We put that in parentheses with R. K. So we get three times the quantity of K plus one plus one, all divided by two in this is the right hand side of our equation for induction with a plus one equal to end

And the given statement is three plus three squared plus three q plus A War plus three. The vehicle to three into three and minus one by Let's All for this one meters equal do three part one with his three and here it is three into three. Part one minus one by this tree is equal to three. Hence the State Ministry. Similarly, let's do it for escape, so that will be in is equal. Okay, so it is three plus three squared plus three q plus one less Staple K will be equal toe three into three Parky minus one. Right, Let's add three into K plus one on board to, say, three in two K plus one onwards. A regular so it will be three plus three squared last three Q plus. So on less freaky last three K plus one that will be equal to 3 to 3. K minus one by two Last three caper This is now so Levi Arches. We will get three in tow. Three Parky minus one bluff doing toe three Parky plus one. By So following this, we will get three into three Park a minus one plus doing +23 Parky taking moment, but no, this will be dream toe three pass key plus one minus one, but no substituting. Yes, K plus one in our e commission, that is a music will do. K Plus one, which will be three plus three squared plus three q plus The war plus three Parky plus one the vehicle Tow three into three Park A plus one minus one by. Does he question in this equation last name and the given Immigration is satisfied and the statement is fully approved.

Okay. To prove this by induction, have to prove our base case to start and equals one left hand side just says the first term of three. And that's gonna be equal to the left hand side three times in which we know to be one times and plus one So one plus one all divided by two. That's pretty easy, cause we just end up having three times one times two divided by two, which just becomes three. So our base case is good. Now we have to prove our inductive n equals K. We know the 369 everything up to three times K is gonna be equal to three k times k plus upon to buy back to. And we're trying to prove the K plus one case. Eso we re ready the left hand side, get our Capel's horn case. We just add three times K plus one. Now that's gonna be equal to a right hand side. Three k times cave plus one Guided by two plus the term we just added three times K one. Your left hand side is the induction step for K plus one. Is that to make it prove the right hand equation for cables More so in order to do that, we're going to I was gonna keep our equation for now, our initial equation for K. And now we're gonna add our second term. But this time we're gonna put it above the denominator of to. So now we have, like, terms, meaning the two in the denominator and the K plus one in the numerator. So by the distributive property, we can get three K plus six times Okay, plus one, all divided by two. So now we factor out of three from our first term three times K plus two times K plus one divided by two. Now this if you look at this closely, it's just the term for K plus One can be written one step further three and then if we switch the terms K plus one in the first spot this time then multiplied by a second term of K plus one plus one, all divided by two. And now we have this remember right here This right here in this case plus one, Is there a new one? So giving k, we proved that this relationship works in the K plus one case


Similar Solved Questions

5 answers
Obtain Eq (2) from the definition of capacitance given in Eq-(l) How are the voltage and distance related? Does In Part 2, this make sense? Explain. why are the voltages - different with and without the material between the plates? And what remains the same, with or without the material?
Obtain Eq (2) from the definition of capacitance given in Eq-(l) How are the voltage and distance related? Does In Part 2, this make sense? Explain. why are the voltages - different with and without the material between the plates? And what remains the same, with or without the material?...
5 answers
5.19Suppose the random variables X and Y have joint pdf f(x,y) = 6y,0 < y < x < 1. Find the marginal pdf of X and marginal pdf of Y_ b. Find the conditional pdf of X given Y = y: Find P(X > #IY = $) d. Find E(X) and E(Y ) Find Var( X) and Var(Y ). f Find Cov(X,Y): g Find the correlation coefficient of X and Y
5.19 Suppose the random variables X and Y have joint pdf f(x,y) = 6y,0 < y < x < 1. Find the marginal pdf of X and marginal pdf of Y_ b. Find the conditional pdf of X given Y = y: Find P(X > #IY = $) d. Find E(X) and E(Y ) Find Var( X) and Var(Y ). f Find Cov(X,Y): g Find the correlation...
5 answers
2ieqHo, HzSOa CCHCH,C -Dnw the structure of the tetrahednl inlcnncdiate INITIALLY FORMED the reiction shoRothiveconsidet stereochctusby-Do not include countcr-ions; € &, Na' , I , in your answtT In cuses whcte therc morc then one OnSWer; juxt dnw one00Chum DcodleaUncnunelotKhnond Exnnmicn6o6
2ieq Ho, HzSOa CCH CH,C - Dnw the structure of the tetrahednl inlcnncdiate INITIALLY FORMED the reiction sho Rothive considet stereochctusby- Do not include countcr-ions; € &, Na' , I , in your answtT In cuses whcte therc morc then one OnSWer; juxt dnw one 00 Chum Dcodl eaUncnunelot K...
5 answers
PhotogatePhotogateAVAVBmAHorizontal TrackmB
Photogate Photogate A VA VB mA Horizontal Track mB...
5 answers
How many moles of sulfate ions are contained in 0.420 moles of aluminum sulfate? 0.140 moles1.26 moles0.840 moles0.420 moles
How many moles of sulfate ions are contained in 0.420 moles of aluminum sulfate? 0.140 moles 1.26 moles 0.840 moles 0.420 moles...
5 answers
-3 9 -10 C~ ~10 ~10~10X=
-3 9 - 10 C~ ~10 ~10 ~10 X=...
5 answers
If $vec{a}=2 x^{2} hat{i}+4 x hat{j}+hat{k}$ and $vec{b}=7 hat{i}-2 hat{j}+x hat{k}$ areinclined at an obtuse angle and $vec{a}$ is parallelto $vec{c}=hat{i}+20 hat{j}+50 hat{k}$ then(1) $x=frac{1}{10}$(2) $x=1$(3) $x=-frac{1}{10}$(4) no such value of $x$ exist
If $vec{a}=2 x^{2} hat{i}+4 x hat{j}+hat{k}$ and $vec{b}=7 hat{i}-2 hat{j}+x hat{k}$ are inclined at an obtuse angle and $vec{a}$ is parallel to $vec{c}=hat{i}+20 hat{j}+50 hat{k}$ then (1) $x=frac{1}{10}$ (2) $x=1$ (3) $x=-frac{1}{10}$ (4) no such value of $x$ exist...
5 answers
Express the equilibrium constant (Kc) for the following reaction (2Z (g) = 3X (g) + 2Y (g (Li 2.5)[XH[YF Kc = [Z12[X] [Y]? Kc = [ZI?[Z]? [XF[YI?Kc[2]? [XP[YIKc
Express the equilibrium constant (Kc) for the following reaction (2Z (g) = 3X (g) + 2Y (g (Li 2.5) [XH[YF Kc = [Z12 [X] [Y]? Kc = [ZI? [Z]? [XF[YI? Kc [2]? [XP[YI Kc...
1 answers
The next day, Kate had 1 cup of cereal ( $15 \mathrm{~g}$ carbohydrate) with skim milk ( $7 \mathrm{~g}$ carbohydrate), 1 banana $(17 \mathrm{~g}$ carbohydrate), and $1 / 2$ cup of orange juice (12 g carbohydrate) for breakfast. a. Has Kate remained within the limit of 45 to $60 \mathrm{~g}$ of carbohydrate? b. Using the energy value of 4 kcal/g for carbohydrate, calculate the total kilocalories from carbohydrates in Kate's breakfast, rounded to the tens place.
The next day, Kate had 1 cup of cereal ( $15 \mathrm{~g}$ carbohydrate) with skim milk ( $7 \mathrm{~g}$ carbohydrate), 1 banana $(17 \mathrm{~g}$ carbohydrate), and $1 / 2$ cup of orange juice (12 g carbohydrate) for breakfast. a. Has Kate remained within the limit of 45 to $60 \mathrm{~g}$ of carb...
3 answers
Let X be the time taken to assemble a car in a certain plant;Xis a random variable having a normal distribution of 30 hours and a standard deviation of 4 hours.Question:What is the probability that a car can be assembled in a period of time greater than 32 hours?
Let X be the time taken to assemble a car in a certain plant; Xis a random variable having a normal distribution of 30 hours and a standard deviation of 4 hours. Question: What is the probability that a car can be assembled in a period of time greater than 32 hours?...
3 answers
Simplify: 2 )Provide your answer below:
Simplify: 2 ) Provide your answer below:...
5 answers
Bonus Question: Verify that the following functions are inverses of each other algebraically:Sr-1 f(r)=- X+5Sr-1 g(*) = x+3
Bonus Question: Verify that the following functions are inverses of each other algebraically: Sr-1 f(r)=- X+5 Sr-1 g(*) = x+3...
4 answers
6. See also Example 2 in Lecture Slides 13) Let V 4 Rn let A e Rnxn with rank n, and let M = (mij) 4 AT Ae Rnxn Define(~): VxV + R(u,v) + (u,v) #u M.v = EEmijuivj i-1j-1 Point) Show that (u, v) equals the dot product of Au and Av_ 6 Points_ Show that ( . qualifies a8 an inner product for V; i.e., show that it is symmetric; linear in the first argument; and positive definite Comment: for proving what property do you need the assumption rank( A) n?
6. See also Example 2 in Lecture Slides 13) Let V 4 Rn let A e Rnxn with rank n, and let M = (mij) 4 AT Ae Rnxn Define (~): VxV + R (u,v) + (u,v) #u M.v = EEmijuivj i-1j-1 Point) Show that (u, v) equals the dot product of Au and Av_ 6 Points_ Show that ( . qualifies a8 an inner product for V; i.e., ...
5 answers
How is the placement of the joints in the pig hind limbdifferent from the joints of the human lower limb? What are some examples of other animals that have jointplacement similar to a pig? What is the evolutionaryadvantage to being built this way?Is it possible for a human (like you!) to imitate this jointconformation? If so, how?
How is the placement of the joints in the pig hind limb different from the joints of the human lower limb? What are some examples of other animals that have joint placement similar to a pig? What is the evolutionary advantage to being built this way? Is it possible for a human (like you!) to imita...
5 answers
Find the indicated IQ score. The graph to the right depicts IQscores of adults, and those scores are normally distributed with amean of 100 and a standard deviation of 15. x=0.1431
Find the indicated IQ score. The graph to the right depicts IQ scores of adults, and those scores are normally distributed with a mean of 100 and a standard deviation of 15. x=0.1431...
5 answers
In a certain city the temperature (in °F) t hours after 9 AM wasmodeled by the function T(t) = 56 − 2.2t + 0.41t^2 − 0.018t^3. Findthe average temperature during the period from 9 AM to 9 PM. (Roundyour answer to one decimal place.)
In a certain city the temperature (in °F) t hours after 9 AM was modeled by the function T(t) = 56 − 2.2t + 0.41t^2 − 0.018t^3. Find the average temperature during the period from 9 AM to 9 PM. (Round your answer to one decimal place.)...
5 answers
Edlniin Hnaealurt Lr CoC ,EuelnlL4 LITIIEA;
Edlniin Hnae alurt Lr CoC , EuelnlL4 LITIIEA;...

-- 0.020636--