5

Starting with a1, write the first four terms of the sequence.an6n! 2n !...

Question

Starting with a1, write the first four terms of the sequence.an6n! 2n !

Starting with a1, write the first four terms of the sequence. an 6n! 2n !



Answers

Write the first four terms of the sequence.
$$a_{n}=\frac{n !}{n^{2}-n-1}$$

So we're going to start this problem. Our problem is a sub in equals in factorial over in squared. And we need to find the first four terms of this sequence. So before we get started, what we want to know is what do these letters mean? So in the letter in is the number in the sequence. So more like a position. So the first number, the second number. The third number. So we're going to say it's the position. That's what I I say a lot of the times because it's easier to remember like oh the first position, the second position, the third in the sequence. So that is like where our number is in the sequence. So is it the first number? Is it the second number on and on? Um and then are a sub in Okay. That we call that because that is a little bit smaller um It's the number in that position for instance. So if we have won our first number, our second number. Our third number in our fourth number. Okay. And are let's say we counted by even numbers and we did 2468. So um We would say in the 4th position a seven is 8. So we would write that as a sub four is equal to eight in this example here. So that's the position and that's the number in that position. The last thing. Just a quick thing with this exact problem. That exclamation point means it's a factorial. All right. So for instance if you had three factorial, that's the same thing as multiplying three times two times one, you just go down in numbers and multiply them. So if we're looking at this exact problem, All right. What we want to know is the first four numbers. So if it helps we can kind of set it up, we need to find the numbers that go in those positions. Um So super easy. All we have to do is plug in these numbers And for in because their position numbers. Okay, so how we set this up as well say a sub one is equal to one factorial over one squared. Ok notice I just took our original problem. Okay I took our original problem and I plugged those position numbers in for our in numbers and now we have to do now is evaluate so we'll say a sub one is equal to well one factorial is just one and one squared is just one. So our fraction now is 1/1. So a sub one is equal to one. And if you want to come up here to our little position and write the number one so we don't forget. That might help a little bit. So that's how we find our first number To find our second number. You know we'll have a sub two is equal to two square or two factorial over two square. All I did was plug in for in so I have a sub two is equal to well two factorial. Don't forget R factorial mean two times one. Because we're going in order we started to we multiply by the lower the next number down. So two times one is two over two squared is four And to over four reduces to be 1/2. And if you want to come up here and right in our little Um spaces that the second number is 1/2. Yeah. All right. I'm gonna give you a second to try to plug in three. So if you want to pause the video you can do that and try it on your own and then I'm gonna do it and we can see how do you do? So you can pause now. All right. So to do our third number again, we'll have a Sub three is equal to three factorial over three squared three factorial is three times 2 times one which is six and three squared three times three is nine and 6/9 reduces to two Over three. So that's our third number is two thirds. And then our last thing we want to do is plugging in four. So I'm gonna erase some of this so I have some space to work. Yeah, it will have a Mhm. Our fourth number is a sub four equals four factorial over four squared four. Factorial is the same thing as four times 3 times two times 1. Which is 24 Over four square to 16. Will reduce that fraction to be 3/2. Okay, I divided both of those by eight and it reduced to three halves. So that's what our fourth number is. So any time you're dealing with a sequence problem, things you want to remember is that the in value is what we plug in for in is the position. So the first number, the second number. The third number of the fourth number. All right. And what it's gonna give us when we plug in is it's going to give us the number in that space, which is a sub N. So all of these mean the number in the first spot. The number in the second spot. The number in the third spot. The number in the fourth spot. All right. And we plug in those physicians to get the numbers in those positions. So, if we um, I want to know what what's the fifth number we plug in five. We want to know what's the 100th number we plug in 100 for in. And we would get, if this pattern continued all the way to 100 we would get what number is in that 100 position. If we continue and continue and continue. So I hope that was helpful. Um all sequences are like this when we can plug in. If we're given a former are a series formula like above.

All right, So for this question, this is just kind of reinforcing that factorial zehr just numbers. They're just values. So keep in mind that we can treat this just like it was a normal end because n factorial is just a value. And this and factorial is gonna have the same value, is this one? So once we go ahead and figure out what these are, if we ever figure out what to sub in for end, these are gonna have the exact same value. So if we're multiplying and dividing by that, it's going to give us the same value no matter what's we can actually cross these out from the top and the bottom, and it's going to give us the exact same answer as if we actually had a value in here for here. It's like if and was actually two on then the value of two factorial was just too. It would just be multiplying 3/4 by two over two, which would just be six eights which would still reduce 2 3/4 So no matter what the value of these two are, our answer's always gonna be 3/4 so we can go ahead and cross those out and just figure out ahead of time that our answer to this factorial is going to be.

Hey, guys. So for this factorial, it's important just to keep in mind that end. Factorial is just kind of going to be a variable. In this case, it's gonna be the exact same as if it was just an end here. So let's go ahead and get started with finding our 1st 4 terms. And then at the end, I'm gonna go ahead and show you a shortcut that's gonna save us a lot of time. If you ever have another one of these questions, you've got one too three and four. Okay, so one factorial is just gonna be one. So we're gonna be left with three times one, which is three over four times one, which is for when we've got to two. Factorial, we already know is too. So we've got three times to its six on four times two is eight, which can actually be reduced to 3/4. So then we've got three factorial three times, two times one is six and then six times three is 18 6 times four is 24 which can take a guess, be reduced to three forts and then we've got four. So that's gonna be four times six, which is 24. And then we're just gonna be left with 4 to 24 times three, which is going Thio give us 72 and then we're gonna have four times 24 which is gonna give us 96 which can be reduced to 3/4. So the way we could have figured this out without doing all of this work is to just keep in mind that these air just like variables. So no matter what happens, we're multiplying both the top and the bottom by an answer, which means this can actually be factored out because this is just gonna keep the same value no matter what. It's just multiplying two things by the same number, so that can always just go ahead and be divided out while keeping the same value. So our answer to all of these and any further numbers in this sequence are going to be

High in this problem. We have to write out the 1st 4 terms of the form and squared over to end. We just get this by substituting each value in tow sequence by substituting one and 4 a.m. We can evaluate these by hand. That's one of the three that's for the five. That's 987 that's 16.


Similar Solved Questions

5 answers
Math SquaresPuzzle 13Puzzle 19ZPuzzle 15Puzzle %6
Math Squares Puzzle 13 Puzzle 1 9Z Puzzle 15 Puzzle %6...
4 answers
M HCI Data: M CH;COOH Part I(c)-Nature Mass of reactants: 3 0 0 7 of MgNumber : IF 40 7u ofMg (mol) Time required (s)
M HCI Data: M CH;COOH Part I(c)-Nature Mass of reactants: 3 0 0 7 of Mg Number : IF 40 7u ofMg (mol) Time required (s)...
4 answers
Ullucalom[Review Topics] [References] Use the References to access important ralues if needed for thisThe illustration to the left represents mxture of renon brown and fluorine green moleculesIf the molecules in the above illustration react to form XeF4 according to the Xe 2F2 XeF4 equationthe limiting reagent 15the nutnber of XeF 4 molecules fored isandthe number ofatoms mlolecules inl excess i5
ullucalom [Review Topics] [References] Use the References to access important ralues if needed for this The illustration to the left represents mxture of renon brown and fluorine green molecules If the molecules in the above illustration react to form XeF4 according to the Xe 2F2 XeF4 equation the l...
5 answers
8) Subiect: ^ male 30 Yr %f age, 450 Ib , Height; 70 inch: VOZmax= treadvallabiadon 2sa and Desired Exercise intensity: 75)MOZReSerVe 6Oml/kg/min; desired 6.3 (available on Blackboard) , determine running speed (Km/h) "Using Table point). on a treadmili (2
8) Subiect: ^ male 30 Yr %f age, 450 Ib , Height; 70 inch: VOZmax= treadvallabiadon 2sa and Desired Exercise intensity: 75)MOZReSerVe 6Oml/kg/min; desired 6.3 (available on Blackboard) , determine running speed (Km/h) "Using Table point). on a treadmili (2...
5 answers
11. F9 Final due Jan 13, 2021 23:00 +08 WBookmark this pageF. (9) 1.0 point possible (graded, results hidden)Evaluate the limit36+ +kn2+n4)_(Enter your answer as a sum of fractions, or as a decimal to 2 decimal places )lim Aet3(++k2,2+7)FORMULA INPUT HELPSubmit
11. F9 Final due Jan 13, 2021 23:00 +08 WBookmark this page F. (9) 1.0 point possible (graded, results hidden) Evaluate the limit 36+ +kn2+n4)_ (Enter your answer as a sum of fractions, or as a decimal to 2 decimal places ) lim Aet 3(++k2,2+7) FORMULA INPUT HELP Submit...
5 answers
Calculate the pH at the equivalence point for the IItration ot 0.230 M methylamine (CH NHz) Wth 0.230 M HCI; The Ku of mathylamlne Is 5.O* 10 4NumberpH =
Calculate the pH at the equivalence point for the IItration ot 0.230 M methylamine (CH NHz) Wth 0.230 M HCI; The Ku of mathylamlne Is 5.O* 10 4 Number pH =...
5 answers
Dmxt4+*.Fennrtltmlha & u Lleltculat {An cIle SXtnE4]ICaulalo r e Ilx (ur @M Fl-I-l-lnel Lt794a nLL
Dmxt 4+*. Fennrtltmlha & u Llelt culat {An cIle SXt nE4]I Caulalo r e Ilx (ur @M Fl-I-l-lnel Lt794a n LL...
5 answers
Solve the initial value problem3x dx + 8x e 8xy dy = 0, Y(2) = 38y e 8xyThe solution is (Type an equation using X and y a5 the variables. Type an implicit solution )
Solve the initial value problem 3x dx + 8x e 8xy dy = 0, Y(2) = 3 8y e 8xy The solution is (Type an equation using X and y a5 the variables. Type an implicit solution )...
4 answers
"Let X, and Xz be independent N(O,1) and x2(r) respectively Consider the following transtonsandY? X[a)] Find the Jacobian of this transformation. [(b)] Find the joint _ distributlon of (Y, Yz)" ((c)] Find the two marginal pdfs of Y and Yi [(d)] Are Y and Yz independent? Justify your answer
"Let X, and Xz be independent N(O,1) and x2(r) respectively Consider the following transtons andY? X [a)] Find the Jacobian of this transformation. [(b)] Find the joint _ distributlon of (Y, Yz)" ((c)] Find the two marginal pdfs of Y and Yi [(d)] Are Y and Yz independent? Justify your ans...
5 answers
What capability does the fstream data type provide that the ifstream and ofstream data types do not?
What capability does the fstream data type provide that the ifstream and ofstream data types do not?...
5 answers
Use Stokes' Iheorem t0 compute D,curl? 4S mtxere F(s,V,2) = Wv- 4+%vk and €C s thue untersecton ol the homusphere ? 132 vath thic plont = Assurnc $ nasan Oumurd noral vector:
Use Stokes' Iheorem t0 compute D,curl? 4S mtxere F(s,V,2) = Wv- 4+%vk and €C s thue untersecton ol the homusphere ? 132 vath thic plont = Assurnc $ nasan Oumurd noral vector:...
1 answers
Use l'Hopital's rule to find the limits in Exercises $7-50$ . $$\lim _{t \rightarrow 1} \frac{t^{3}-1}{4 t^{3}-t-3}$$
Use l'Hopital's rule to find the limits in Exercises $7-50$ . $$\lim _{t \rightarrow 1} \frac{t^{3}-1}{4 t^{3}-t-3}$$...
5 answers
Attach and File 1 5 sec? seconds. are the The field conducting HV loop Browse lies S the 1 the 412 emf +2t magnctic M With field Eccion that is the Teslasted loop 5 1
Attach and File 1 5 sec? seconds. are the The field conducting HV loop Browse lies S the 1 the 412 emf +2t magnctic M With field Eccion that is the Teslasted loop 5 1...
5 answers
The acceleration of an object (in m/s ) is given by the function a(t) J sin(t) The initial velocity of the object is "(0) nV S: Round Vomr answers t0 four decimal placesa) Find an equation v(t) for the object velocity:v(t) Previcw Snx CrTOF Entchan Alesbrak @prersio2 Wn b) Find the object's displacement (in meters) from time to tImcPreviewmetetsc) Find the total distance traveled by the object from time Lo timCPrevicwTclen
The acceleration of an object (in m/s ) is given by the function a(t) J sin(t) The initial velocity of the object is "(0) nV S: Round Vomr answers t0 four decimal places a) Find an equation v(t) for the object velocity: v(t) Previcw Snx CrTOF Entchan Alesbrak @prersio2 Wn b) Find the object&#x...
5 answers
Problem 4* (Optional for extra credit): Let {xn}iz1 be a sequence such thatlimn inf €n = 0. Tn > 0. Vn € N.Prove that there exists subsequence {"ux}k-1 such that(Vk: € N)(Vn € N)n < nk = In 2 Euk-
Problem 4* (Optional for extra credit): Let {xn}iz1 be a sequence such that limn inf €n = 0. Tn > 0. Vn € N. Prove that there exists subsequence {"ux}k-1 such that (Vk: € N)(Vn € N)n < nk = In 2 Euk-...
5 answers
To answer the questions, interpret the following Lewis diagram for NO}1. For the central nitrogen atom: The number of non bonding electrons The number of bonding electrons The total number of electrons2.The central nitrogen atom obeys the octet rule_ B. has ess than an octet: C:has more than an octet:
To answer the questions, interpret the following Lewis diagram for NO} 1. For the central nitrogen atom: The number of non bonding electrons The number of bonding electrons The total number of electrons 2.The central nitrogen atom obeys the octet rule_ B. has ess than an octet: C:has more than an oc...

-- 0.020889--