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Look at the stair shape drawn on the 10 by 10 Number Grid below91 92 93 94 95 | 96 97 98 99 |10081 82 83 84 85 86 87 88 89 90 71 | 72 | 73 | 74 /75 | 76 | 77 | 78 7...

Question

Look at the stair shape drawn on the 10 by 10 Number Grid below91 92 93 94 95 | 96 97 98 99 |10081 82 83 84 85 86 87 88 89 90 71 | 72 | 73 | 74 /75 | 76 | 77 | 78 79 8061 | 62 | 63 64 | 65 66 67 68 69 7051 | 52 | 53 | 54| 55| 56 57 58 59 6041 | 42 | 43 | 44 | 45 | 46 47 48 49 5031 | 32 | 33 | 34 | 35 / 36 37 38 39 4021 22 | 23 | 24 /25 / 26 27 28 29 3011 | 12 | 13 | 14 | 15 / 16 | 17 | 18 19 20 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 10This is a 3-step stair.The total of the numbers inside the stair s

Look at the stair shape drawn on the 10 by 10 Number Grid below 91 92 93 94 95 | 96 97 98 99 |100 81 82 83 84 85 86 87 88 89 90 71 | 72 | 73 | 74 /75 | 76 | 77 | 78 79 80 61 | 62 | 63 64 | 65 66 67 68 69 70 51 | 52 | 53 | 54| 55| 56 57 58 59 60 41 | 42 | 43 | 44 | 45 | 46 47 48 49 50 31 | 32 | 33 | 34 | 35 / 36 37 38 39 40 21 22 | 23 | 24 /25 / 26 27 28 29 30 11 | 12 | 13 | 14 | 15 / 16 | 17 | 18 19 20 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 10 This is a 3-step stair. The total of the numbers inside the stair shape is 25 + 26 + 27 + 35 + 36 + 45 = 194 The stair total for this 3-step stair is 194. Part 1 For other 3-step stairs, investigate the relationship between the stair total and the position of the stair shape on the grid. Part 2 Investigate further the relationship between the stair totals and other step stairs on other number grids.



Answers

Staircase Problem: Debbie can take the steps of a staircase 1 at a time or 2 at a time. She wants to find out how many different ways she can go up staircases with different numbers of steps. She realizes that there is one way she can go up a staircase of 1 step and two ways she can go up a staircase of 2 steps $(1 \text { and } 1$ or both steps simultaneously). a. Explain why the number of ways she can go up 3 - and 4 -step staircases are 3 and 5 respectively. b. If Debbie wants to get to the 14 th step of a staircase, she can reach it either by taking 1 step from the 13 th step or 2 steps from the 12 th step. So the number of ways to get to step 14 is the number of ways to get to step 13 plus the number of ways to get to step $12 .$ Let $n$ be the number of steps in the staircase, and let $t_{n}$ be the number of different ways she can go up that staircase. Write a recursion formula for $t_{n}$ as a function of $t_{n-1}$ and $t_{n-2} .$ Use the recursion formula to find the number of ways she could go up a 20 -step staircase. On your grapher, you must enter $u(n \text { Min })=\{2,1\}$ to show that $t_{2}=2$ and $t_{1}=1$ c. How does the number of ways of climbing stairs relate to the Fibonacci sequence in Problem $17 ?$ d. In how many different ways could Debbic go up the 91 steps to the top of the pyramid in Chichen Itea, Mexico? Surprising? PICTURE CANT COPY

So in this problem we are given Fibonacci is famous sequence as I know that at the top. So in part a of this problem, we want to figure out what would be the recursive formula to find any term of the sequence. Well, the cool thing about this formula is that after the first two terms to get to the next term, you add the previous two. So in other words, to find the third term ace of three, we're going to add the 1st and 2nd terms one plus one. That would be too. So to find the fourth term, we're going to add the previous two terms. No, this one plus two is equal to three. So like to find the fourth term ace of five, we would add this 3rd and 4th terms two plus three, which is equal to five. And you'll notice this pattern throughout. So the way we can Right, right. And the formula Right and recursive formula is that we would say a seven. Any term of the sequence is found by taking the two previous terms and adding them together. So we take a sub minus two. That's the term to before it. And we're going to add a sub minus one, that's the previous term. But we have to establish that. So we have to say when a someone has to equal one and a sub two has to equal to two. Because if we had values of end that were less than two, that were either one or two, we make our subscript negative and we wouldn't have a term for those. So this would be a recursive formula. Now the second part of this problem wants you to go to your calculator to enter in uh the formula and then you're going to scroll down to find the 20th term. So what you should find is that that 20th term ace of 20 would be equal to 6,765. So that would be the 20th term of this fibonacci sequence. Okay, so now let's take a look at heart view this problem. What we're being asked to do is to find the 1st 10 ratios and all that means is we're going to take two terms and divide them. So the first one we divide the second from the first. So the second term is one. The first term is one and one divided by one is one. to find the 2nd 1. We're going to take the third term which is to Divided by the second term which is 1, 2 divided by one is 2. All right. So now the next one we're going to divide three x 2. So in this case because we divided by one. Number one, we did to buy one. We're gonna divide three by two. I just wanted to make sure and three divided by two is equal to 1.5. All right Then we're going to divide five x 3. So we'll go back to our list here. Five divided by three. That would be equal to 1.6 repeating. Alright, we go back to our list now we're gonna divide eight x 5 so we'll scroll down eight divided by five and that was equal to 1.6. All right. Next on our list, 13 divided by eight. So 13 divided by eight and that was equal to 1.625. So notice how our values are getting closer and closer to a specific number. All right. Our next 121 divided by 13. So actually I'll go here so I have to scroll as much. So 21 divided by 13, that would be equal to 1.615. I'll just go three places after the decimal. Next 34 divided by 21 234 divided by 21 which would be equal to approximately 1.619. Next we have 55, divided by 34 And 55 divided by 34 would be 1.617. And how many have we done here? It looks like we've done 123456789 We have 21 more. So we have to find the next term of the sequence. So we add 34 55. That would give us 89. So now we're going to divide 89 x 55. So 89 divided by 55 which is equal to approximately 1.618 without 18 just keeps repeated. So what we wanted to know this is that these values are getting closer to that golden ratio of 1.618. And that's exactly what's happening in this case. And that would prove it now for parts in this problem. They want you to look up at pine cone, a pineapple or sunflower and they want you to look at the cross sections and those the numbers and their spirals. And what you'll find is that as you go out, the numbers actually form are fibonacci sequence pattern. So this is how Fibonacci numbers can be used in a lot of different ways. And in part d what they want you to do is to look up some information about Fibonacci and figure out where he lived and all of that stuff. Um And to kind of look up the population growth for rabbits. And again, you'll find that it actually kind of follows from that pattern. So that's why the sequence can be used in multiple ways.

Hey, for this problem, we're going to be examining the positions of people in a marching band. We're going to do this in a few steps. First, we have, uh, matrix that they've given that they're calling B. And as you can see, if you look at how be a set up, the third row is just once. There's one person in each place, but the top and the middle rows 1st and 2nd Rose show the positions of three different band members. If you kind of think of the football field is a grid, you've got one band member at 50 0 one at 50 15 and one at 45 23 non ca linear band members and we've been asked to start with We're gonna find the inverse of B. So if this is be that we're showing here, I've got a copy to be on the screen to find the inverse. We're going to remove that second bracket and we're going to augment it. We're on the let right hand side. We're going to copy the identity matrix, and we're going to use row manipulation to make the left hand side look like the identity matrix and when that's done, the right hand side will be the inverse. So in order to do this, the first step, we're gonna leave the first row exactly the way it iss, huh? And my goal is, since the identity matrix has an element in the first row. First column will leave that first row first column just as it is. And they will make everything else in that first column equal to zero. Well, one thing that makes our job a little easier. There's already a zero in the second row so I can leave those numbers alone. I don't have to do any manipulation on those, but what about the third row? I do have a one here, and I need to change it to a zero. So my row manipulation will be to take the opposite of the first row, plus 50 times the third row. And those will be my new third row entries all the way across. And when I do that, that's going to give me a new third row of 005 Negative 10 50. Hey, first row is done, or the first column now the second column. There is an element in the second row. Second column position of the identity, Uh, matrix. So I'm gonna leave the second row alone. Copy it just the way it is. And I want to have zeros everywhere Else in the second column again, we got kind of lucky. The third row already has a zero there, so no manipulation needed. We could just copy it. But what about the first row? I have a 50 here that I need to get rid off. So what I'm going to end up doing is I'm going to take negative 10 times the second row, and I'm gonna add three times the first row. Doing that gives me a new top row. And my new top row is going to be 150 0 negative. 65 three negative. 10 0. Hey, one Maurin oration should have all of our zeros in the proper place. Uh huh. There is a There is an element in the third row. Third column position of the identity matrix. So we're gonna leave the third row just the way it is and deal with that third column. Our goal is to make every element in the third column except for that third row third column spot to equal zero. So what do I need to dio? Well, first of all, let's look at the first row. I have a negative 65 so I'm gonna take 13 times the third row and add it to the first wrote. Doing that gives me a new top row. 150 00 negative. 10 Negative. 10 650. Hey, now, my second row, I have a 20 that I need to get rid of. So I will take the opposite of four times the third row, and I'll add that to the second row. That gives me a new second row of 0. 15 0 41 negative. 200. Okay, we are just about done. I'm going to just scroll a little, give us a little bit of space. It almost looks right. All the zeros in the right spot. I just need ones on that diagonal. So I'm going to divide everything in the top row by 150 everything in the second row by 15 and everything in the third row by five. And that will give me my new inverse matrix. So the left hand side is now the identity matrix and the right hand side is negative. 1/15 Negative. 1/15 13 3rd. Second row is 4/15 1/15 Negative, 43rd. And the bottom row is negative. 1/5 zero and 10. So this is the inverse of matrix B. Great. So we're almost ready to find our new band positions. But first, we need to compare this be have to multiply it with our grid. A and A is going to have the positions to which they will be moving. So this is gonna be a multiplication. I've got a times the inverse of be so first before we look at where it comes from. Let me just re copy this inverse matrix. You need to be able to see that top row. So it's negative. 1/15 Negative. 1/15 13 3rd for 15th, 1/15 Negative. 43rd negative. 1/5 0 10. Okay, just re copy. So a is our new positions. So the first column of B was the person at 50 0 at that point on the grid, and we're told that they were moving to the 00.0.40 10 and again we're going to have ones across the bottom. The person represented by column to the one at 50 15 is now moving to 55 10 and the third person who is at 45 20 is moving to 60 15. Okay, so we need to multiply this out. And once we do this will give us our movement matrix. It'll take where they are now and show us where they're going to be. So when we do our matrix multiplication, just a reminder. Here we take the first row times the first column, and that becomes our first row first column entry. So 40 times negative 1/15 plus 55 times for 15th, plus 60 times negative, 1/5. And when we do that, we get an entry of zero. Hey, second row sec. Oh, sorry. First row, second column that will give us our first row second column value doing that. We get 40 times negative 1/15 plus 55 times 1/15 plus 60 times zero. And that gives us a value of one. And we're going to do that for all nine, uh, entries. All the rows and all the columns and our result is 01 40 Negative. 10 60 and 001 Oh, on I apologize. I labeled this wrong. That is not a Our result is a They did not actually name this first matrix for us. I miss I miss read where that a was. My apologies on that. So this is a This is our movement matrix where you start from toe where you go. This matrix will help you determine that. So let's try it with some of the other. We had nine players on the field. We've already done three. So let's look at some of our other players. The one who is furthest on the left is starting at the 10.40 20 on the grid. So what's his new position? Well, we're gonna take that matrix a that we just found, and we're going to multiply it by a matrix that shows where this player started. They started at 40 20. We're gonna put a one in that third position, and when I multiply that out, I'm just gonna put this I'm gonna just put a little thing here over on the side. I'm gonna put our matrices to show where are new positions are multiplying. First row by first column. Gives me 60 second row by that column is 20 and then I get a one in my third position. So the person who starts at 40 20 moves to the position 60 20. What about the next person? Well, in order to do that, all we have to do is erase the current position that we're looking at. Put in the new one. The next person. Next Thio. Next person over. I'm just gonna go along all five of those at the top. The next one is one of the ones we already looked at, so I can ignore him. The next one over is going to be at 50 20. And when I multiply that out, this new person is going to end up at the 0.60 10. Okay, next one over. I could just erase these numbers. This next person is at 55. 20. Doing that multiplication gives me a final spot of 65. Okay, 60 across, five up. Hey, next person is at 60. I'm sorry. It's It's 60. 20 60 20. Multiplying that out gives me a new position of 60 0. Okay. Two more to go now. I'm gonna come down that t it's on the grid that we have. The next person that we haven't looked at yet is standing at the 0.50 10. And actually, when we do that multiplication, this person doesn't move. There must be a pivot point. So they stay at 50 10. No motion. And our last person is at 55 and they end up at the 0.45 10. Okay, so those air, all of our players Now, what does our final shape look like? Well, I took all nine points, the six over here with the Red Star, plus the three that we were given within the problem, and I've graphed them onto a graphic application. As you can see, we're still in a T, but it's gone sideways. It's fallen over to the right. So this tea is our new position. Based on the information given in the problem

We're going to do problem number 43 in this question or we have to Do we have to just find the length off hand? Real. Okay, So it is said that first of all, let me draw the diagram just to make you better understand. Okay, total numbers are 1234 So one, 23 and four. Okay, this is the given conditions in the book on it just said that the handrail is handle is as long as the distance between the edge of first this tape, this question state on the distance between the top page. Okay, so total total part. As as this point, we will count like this that they started parts. Okay, So 123 and four, one toe, three and four means we're getting for four handles. Okay, We had we had just bringing them just to get the length off each steps. Henry, look, it steps Hendry as we are given this, that is, this is Children. This is seven. The dimension off each step is given. This is challenges. And this is seven inches, so we can find the length off each Hendren off each step. Then we will just multiply that with four, as we have to find still fourth step. 123 and four. The step So we'll just find the length off. Lend off question real one step 100. So this is Children. This is seven given, so and it is 90 degree. So by using category still, American diet have renaissance quality goes toe basis carp outside the square. So I put any wear to find there. There's a story square. How do you see in the square? So we're getting happiness Risk articles to 144 plus 49. So I pretended physicals Underwood off. Nine plus four is 34 plus, uh, 48 plus one is nine and 193. This is equals to 13.89 inches. Okay, this is the length of hyper tennis. Now, to get to get the length off handle, we have to multiply. It multiplied with four. Henry. Liz, this is Hendel 13.89 into four. We will find this quantity. That car dean point eight line into four. So this is equals to 55 0.56 And okay, so this is the answer. Thank you. Very


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