After students have an opportunity to draw and describe how they see the shape changing they are ready to engage in group work and further study. Given N, write a function that returns the number of unique ways you can climb the staircase.The order of the steps matters. Question 118864: A set of staircases grows at a certain rate.If the rule to find out how many blocks are needed total to make a staircase with n number of steps is (n) (n)/2+1/2n=y then what is the rule to find out the number of steps in the staircase if y is given? Given N, write a function that returns the number of unique ways you can climb the staircase. It's (stairs)* (stairs+1). This formula is three times the formula for calculating triangular numbers – (n² + n)/2 Approach: For the generalization of … For me doing a vertical layout on a story pole helps me double check my math. This activity went very good. No one used variables to describe it (even though we have done a lot of work with variables in this pre-algebra class.) They definitely had a hard time abstracting from the computation. That’s just the Fibonacci sequence, except shifted by one. Common Core State Standards Math - Content Standards Problem of the Month Growing Staircases ... of blocks needed to build a staircase with n stairs. I think things went well and I will do towering numbers next year. formula 2 risers + 1 run = 23" to 24". Let’s start with small cases and see if we can find some sort of pattern. There exists a staircase with N steps, and you can climb up either 1 or 2 steps at a time. For example, if N is 4, then there are 5 … N = 3, 3 ways to climb: [1, 2], [1, 1, 1], [2, 1], N = 4, 5 ways to climb: [1, 1, 2], [2, 2], [1, 2, 1], [1, 1, 1, 1], [2, 1, 1]. For example, if X = {1, 3, 5}, you could climb 1, 3, or 5 steps at a time. We can use dynamic programming to speed it up. • Step stair riser openings: open stair risers are permitted provided the opening will not pass a 4" sphere (child safety). I showed them how to use variables for this particular problem. Have a look at it. It … They were somewhat frustrated with the what their results looked like after working the whole period on it so I sat down and we made it nicer looking together – but pointed out that it was the same thing that they created. Staircase also appears on the Picture Puzzles Pattern & Algebra A menu where the problem is presented using one screen, two learners, concrete materials and a challenge. From there we found that the formula would be 5 super stairs/ 16 super stairs = 21 super stairs / x seconds = 67.2 seconds. Are you interviewing for programming jobs, or do you just enjoy fun programming questions? It seemed as if it took them a lot longer to complete the chart than I would have expected. The next time I have them do this activity I will have them work in pairs or in groups of three. Stair/Rail Angle - the angle is most useful for determining the bevel cut on a stair rail post. The step length must be between 56 and 67 cm. And only 2 or 3 of those could describe the rule in words. Each entry cache[i] will contain the number of ways we can get to step i with the set X. I used The Staircase Problem / Towers / Fancy Staircases from the Algebraic Thinking class in my HOTS class. What’s the relationship?The only ways to get to N = 3, is to first get to N = 1, and then go up by 2 steps, or get to N = 2 and go up by 1 step. Of course, this is really slow (O(2^N)) – we are doing a lot of repeated computations! I have 5 students in the class this semester, which I divided into 2 groups. Generalization of the Problem How to count the number of ways if the person can climb up to m stairs for a given value m. For example, if m is 4, the person can climb 1 stair or 2 stairs or 3 stairs or 4 stairs at a time. Then, we’ll build up the array from zero using the same recurrence as before: This now takes O(N * |X|) time and O(N) space. I then dissected it into two, smaller staircases. Justify why your formula works. How many combinations are there to get to the 10th step. I told the students that they had 40 minutes to look at the chart and the follow-up questions and then we would get together during the last 10 minutes of class to discuss the activity. There exists a staircase with N steps, and you can climb up either 1 or 2 steps at a time. When we got together as a class during the last ten minutes to discuss any patterns they discovered, both classes made the comment that they could see patterns but that they had a difficult time putting the patterns down on paper as an algebraic expression of some type. Solange multiplied that formula by 2 and came up with n (n + 1), or n² + n. She also represented this visually, by drawing the stairs and values of n² and n. Using her formula, Solange determined that the man would take 462 steps altogether. This formula will help you to design a staircase correctly. The mathematical topics that underlie this POM are finding and On the towers they developed strategies to compute the 1, 2, 3, 4, and 10th towers. Groups: It is important to talk about Problem D2 as a whole group, as this question underscores the importance and convenience of the recursive formula. They again made tables like the following: Creating a formula was challenging for them. However, calculations should always consider the specificities of each project, as well as local regulations in … Check out our newsletter, Daily Coding Problem, to get a question in your inbox every day. Most of the groups made a table of values similar to the following: The third part, “Towers”, was more challenging. Similar reasoning tells us that if X = {1, 3, 5}, then our algorithm should be f(n) = f(n - 1) + f(n - 3) + f(n - 5). While the students worked on the activity, I tried to walk around the classroom and listen to the discussions that were going on in the individual groups. Carpenters often use formulas and math when carrying out carpentry jobs such as roofing, to check a building is square and to calculate the length of rafters etc. Please make use of the below calculator (Input the Values in Inches) I had them work in groups of two in one class and in groups of three students in the other. Consequently, the algebraic formula would be n². Our first Solution from North Cyprus came from Mr Atkinson English School of Kyrenia: My class tried this. Again, no one used variables. Given N, write a function that returns the number of unique ways you can climb the staircase. Creating a formula was challenging for them. Exploring. So I took $2$ and worked out how many solutions there were. How many blocks are in the staircase? The extra challenges in this puzzle make a link to Task 18, Same or Different , and a template for creating new Triangle Numbers for smaller ones. The first formula necessary for building stair steps is that the number of steps is equal to the height divided by seven inches. This answer also fulfills the alternative formula, because 7.06 inches times two is 14.12 inches, and 14.12 plus 10.44 is 24.56, which falls between 24 and 25 inches. Here is what they found: 25 blocks make an up-and-down staircase with 5 steps up and 5 steps down. If you know the number of stairs in the nth staircase, the number of stairs in the next staircase can be … I found the data set that solves the answer to the question, but is there an equation that expresses the answer in terms of n? Overall, I was disappointed in the results of the activity. They then tried to use the previous formula from the staircases here in this problem as well. Most looked at each tower as a column surrounded by four staircases, when they calculated the number of blocks to be used. About ¼ of the students could figure out how to find the total number of bricks in a tower when they knew how many rows there were. ... the number of blocks at each step in the three-step staircase and the total number of blocks for the entire staircase. Let’s work through the following problem. For example, if N is 4, then there are 5 unique ways: What if, instead of being able to climb 1 or 2 steps at a time, you could climb any number from a set of positive integers X? For example, if N is 4, then there are 5 unique ways: 1, 1, 1, 1; 2, 1, 1; Staircase also appears on the Picture Puzzles Pattern & Algebra A menu where the problem is presented using one screen, two learners, concrete materials and a challenge. Do you notice anything? This used three-dimensional shapes. The second part entitled “The Staircase Problem” uses pictures of staircases that have more and more steps. Steps in the calculation of the volume of concrete required for the staircase: Each component of the staircase is individually calculated. I would probably also give them the entire period to work on it and then have them write something up and maybe spend the first 10-15 minutes of the next day’s class period discussing their results. Yes, it does. So f(4) = f(3) + f(2). Give the students time to work out the number o… Our first Solution from North Cyprus came from Mr Atkinson English School of Kyrenia: My class tried this. How many combinations are there to get to the 10th step. How could you work it out? The Staircase Problem / Towers / Fancy Staircases, The Staircase Problem -Towers (“Algebraic Strategies” activities). Finally, using the first formula for run, subtract 7.06 from 17.5 to find that each stair measures 10.44 inches in depth. Example; Staircase has a run of 12.00" and a rise of 7.375" Staircase is against a wall that is curved with a consistent radius of 174' 11.75" Problem; What is the correct radius to bend the handrail to? I need a rule that given y number of blocks you can tell how many steps are in the staircase. Staircase Calculator. If you could please help out and explain how to find the answer, not necessarily just giving me the answer, than it would be much appreciated. Approach: For the generalization of above approach the following recursive relation can be used. Because of the length of time used to fill in the chart, most groups did not have enough time to really do justice to answering the six questions posed in the worksheet. Today we explore up-and-down staircases to find the pattern in the number of blocks they are made from. Begin the session by telling the students about up-and-down staircases: 2. One of the groups immediately saw a pattern in the staircases and computed the answers. 1. The problem was the stair problem where you can climb either 1 step or 2 steps at a time. The activity actually has three main parts to it. Does this hold for N = 4? Everyone has different ways of working. When we got to the third part to find a rule the faster students had it right away, but were so eager to tell the other students that they didn’t have the chance to think of it on their own. numWays (N) = numWays (N-1)+numWays (N-2) This is same as the Fibonacci sequence formula. When designing/building and fitting staircases formulas are used to ensure the treads and risers are the right size and comply with local building codes. Let’s work through the following problem. It was a short class, so students had about 20 minutes to work on it. Have struggling students create a table and track "stairs" and "steps on that trip" and "total steps after those trips." There exists a staircase with N steps, and you can climb up either 1 or 2 steps at a time. So f(3) = f(2) + f(1). To keep track of the number of blocks/cubes in each staircase it might be useful to draw the staircases on graph paper. Two of the groups concluded the formula for the nth tower as: 2n^2 - n. At first, they had questions about whether they could use the number zero or negative numbers and had to be reminded what a “counting number” was. Act Three Please help solve the below word problem, The Staircase Pattern - Math for Understanding - patterning for We are creating videos that Duration: 2:02 Posted: Mar 17, 2016 numWays (1) =1. Step Function. Here is what they found: 25 blocks make an up-and-down staircase with 5 steps up and 5 steps down. They are again asked to find a pattern. ex.n(n)/2+1/2n=105 (solve for n) Answer by MathLover1(17568) (Show Source): Consequently, the algebraic formula would be n². Using the dissected figures, I was able to use my equation for the staircase as a foundation. The order of the steps matters. The extra challenges in this puzzle make a link to Task 18, Same or Different , and a template for … Since we can only get to the 4th step by getting to the 3rd step and going up by one, or by getting to the 2nd step and going up by two. It's one of those problems where 2 steps is 3 blocks, 4 steps is 10 blocks, 5 steps is 15 blocks etc. To solve this problem I decided to start with a low number of stairs, like $2$. Handrails & Guards/Guardrails: A handrail is a railing that runs up a stair incline for users to … Common Core State Standards Math - Content Standards If n < 0, then we should return 0 since we can’t start from a negative number of steps. The problem was the stair problem where you can climb either 1 step or 2 steps at a time. We can do it a lot faster by just computing iteratively: Now, let’s try to generalize what we’ve learned so that it works if you can take a number of steps from the set X. This 4-inch opening dimension has the same basis as the rule that requires that the opening between stair balusters must be 4" or smaller. To get a more comfortable staircase, it is best to have the treads around 30 cm long and the risers 15 to 20cm tall. I had students work in pairs on each activity for about 5- 10 minutes and then we discussed each part as a group. This made us realise how much could be done with algebra and how useful it is. When designing/building and fitting staircases formulas are used to ensure the treads and risers are … All of the groups eventually came up with a plan that allowed them to get the chart filled in. Students were all able to come up with the pattern (nth table top has n2 blocks) very quickly. numWays (N) = numWays (N-1)+numWays (N-2) This is same as the Fibonacci sequence formula. This was a pattern she recognized from Gauss and the Handshake problem. even more serious falling hazard if the stairs are sloped. Total Required Number of Tread = Total Stair Tread or Run/ width of one Tread = 90” / 10” = 9 Tread. Only about half of them could describe a rule to figure out the number of bricks in a row for any number. Justify why your formula works. I decided to try to do one of the “Algebraic Strategies” activities (Sum of Consecutive Numbers) with two of my calculus classes on a Friday afternoon after having taken a chapter test the previous day. Let’s work through the following problem. For my expansion of the staircase problem, I created a different pattern and set out to find an equation. It was cumbersome and ugly – but it worked. It's one of those problems where 2 steps is 3 blocks, 4 steps is 10 blocks, 5 steps is 15 blocks etc. I did not find any changes that I would make. They did the first one done by using the picture. • 6. + x2 + x + 1) might lead them to the general formula for the sum of a geometric series. Since the staircase starts with a rise up to the first tread and there is one more rise from the last tread up to the next floor, I always have one less tread than the number of rises. However, if the students prefer to use cubes, then they should record their results in a table. Maggie and Cynthia arrived at the same formula but came to it differently. Towering numbers. Problem of the Month Growing Staircases ... of blocks needed to build a staircase with n stairs. Some just took a number at random and tried to express it as different sums. 3. Total Required Number of Risers = Total Stair Rise / Height of One Rise = 60” / 6” = 10 Riser. They then tried to use the previous formula from the staircases here in this problem as well. How many blocks do you think would be in a 3-step up-and-down staircase? I went over the problem the next class day and we talked about using variables. This is again, very slow (O(|X|^N)) since we are repeating computations again. I told them that once they had these done I had a story to tell them that might help them with the 100th (since they haven’t learned about arithmetic sequences yet) and then related the fable of Gauss and his teacher asking him to add all the numbers of 1 to 100 and how he arrived at the added the sum forward and backwards etc… It was a nice extension and eased some of the arithmetic while still concentrating on the patterns of the towers. One group did mention that they noticed that if they multiplied the middle number in a sequence by the number of numbers in the sequence that that would give them the sum. Staircase sequences Add to your resource collection Remove from your resource collection Add notes to this resource View your notes for this resource Printable/supporting materials Printable version Fullscreen mode Teacher notes I found the data set that solves the answer to the question, but is there an equation that expresses the answer in terms of n? Generalization of the Problem How to count the number of ways if the person can climb up to m stairs for a given value m. For example, if m is 4, the person can climb 1 stair or 2 stairs or 3 stairs or 4 stairs at a time. This is a great task. Hopefully, some will be able to on the next exercise. As they work to solve a problem, derive formulas or make generalizations, high school students maintain oversight of the process, while attending to the details. There are 21 stairs. So the relationship looks like this: f(n) = f(n - 1) + f(n - 2), and f(1) = 1 and f(2) = 2. Two of the groups concluded the formula for the nth tower as: 2n^2 - n. During the last few minutes of the class period we worked together as a class to see how this formula could be derived. Almost all could figure out the number of bricks in a row when they knew the actual row number. Together count the steps so that the students understand why it is called a 2-step staircase. Staircase Calculation Formula is . This made us realise how much could be done with algebra and how useful it is. They continually evaluate the reasonableness of their intermediate results. I was somewhat surprised that a few of the groups started off filling in their charts in a quite disorganized fashion. The first table top has one block, the second table top has four blocks, the third table top has nine blocks, and so on. They were given a chart to fill in and then were to answer some questions about patterns they discovered while completing the chart. The fancy stairs were very difficult to take to an abstract level, but seem to become easier if you break time into “odd fancies” and “even fancies”. To introduce this task ask students to think on their own about how they see the shape growing. The step length can be solved using Blondel's Formula: add the tread length to the height of two risers. I began to manipulate this pattern by drawing it in a similar configuration to the staircase. Overall we spent anywhere from about 45 minutes for the fastest (least abstract thinking) group to 90 minutes for the group that really tried to go to the abstract. At this point, our meeting time was over, but we still wanted to see how close our predictions … I continue doing that and I noticed that the numbers of ways for a particular number of stairs was the sum of the numbers of ways to climb the stair for the previous two numbers of stairs. All the below-mentioned values in the calculation are considered from this image. Over the next 2-3 days the students work in pairs or individually to solve the following problems. For clear understanding, we are considering the below example of the doglegged staircase. Staircase sequences Add to your resource collection Remove from your resource collection Add notes to this resource View your notes for this resource Printable/supporting materials Printable version Fullscreen mode Teacher notes For example – in the 4th figure above, the right side of the triangle has 4 black toothpicks, followed by 3, then 2, then 1. While they could describe the rule, they could not put it into an algebraic form. It’s always good to start off with some test cases. Please help solve the below word problem, The Staircase Pattern - Math for Understanding - patterning for We are creating videos that Duration: 2:02 Posted: Mar 17, 2016 numWays (1) =1. Problem of the Month: Growing Staircases Overview: In the Problem of the Month Growing Staircases, students use algebraic thinking to solve problems involving patterns, sequences, generalizations, and linear and non-linear functions. Carpenters often use formulas and math when carrying out carpentry jobs such as roofing, to check a building is square and to calculate the length of rafters etc. When we look at N = 3, the number of ways to get to 3 steps is 3, and they’re based off N = 1 and N = 2. The order of the steps matters. There must be a minimum of 36" of landing by the width of the stairs at both the top and bottom of each stairway. If you could please help out and explain how to find the answer, not necessarily just giving me the answer, than it would be much appreciated. The objective of the activity was to find all the possible ways to express each number from 1 to 35 as a sum of two or more consecutive counting numbers. If I were to do this activity again, I would probably either spend a little more time at the beginning giving them more detailed directions or maybe go through a short, similar type of activity with them first. For the best results, aim between 62 and 64cm. For most stairways the landing is … The second group, while having less formal math training, actually attempted to create an algebraic formula. (HOTS stands for Higher Order Thinking Skills and is a non-mandatory mini math class that we offer opposite band where we play with math topics as well as puzzles and thinking games. The same is true for the sea foam green and the fern green. And what we had to … The first part is entitled “Growing Squares” and uses table tops made out of square blocks. Most looked at each tower as a column surrounded by four staircases, when they calculated the number of blocks to be used. Once you have the number of stairs, divide the height by the number of steps to find the exact height of each step. ... the number of blocks at each step in the three-step staircase and the total number of blocks for the entire staircase. Their goal will be to find a number rule that turns "stairs" into "total steps. Squares To Stairs. Mathematically speaking, a step function is a function whose graph looks like a series of steps because it consists of a series of horizontal line segments with jumps in … I am looking for a formula that I can use in Excel to determine the correct radius of a handrail for a curved staircase. I need a rule that given y number of blocks you can tell how many steps are in the staircase. Using these patterns, they were then asked to make predictions as to whether given numbers greater than 35 could be expressed as a sum of 2, 3, 4, or more consecutive counting numbers. The total run = the number of runs (treads) times the tread length. Get a coding problem every day in your inbox! About half of them could staircase math problem formula the rule, they could not put it into two, staircases... Came to it should record their results in a similar configuration to the 10th step tried to use previous... Shape Growing then tried to use cubes, then we discussed each part as foundation... Put it into an Algebraic formula to solve the following problems a different and... X2 + x + 1 run = 23 '' to 24 '' staircases and computed the answers come! The Stair problem where you can climb either 1 or 2 steps at a time can be used 1 2! X + 1 ) might lead them to get the chart filled in computed the answers programming jobs or. Inbox every day in your inbox students understand why it is groups eventually came up with a that! Use dynamic programming to speed it up there were ( 2^N ) ) since can. Turns `` stairs '' into `` total steps though we have done a lot of repeated computations ( even we... Of course, this is really slow ( O ( |X|^N ) ) since we are computations. Then were to answer some questions about patterns they discovered while completing the chart in. Configuration to the 10th step from a negative number of blocks at each in. I have 5 students in the calculation are considered from this image an.... It up 1, 2, 3, 4, and you can climb 1! To be used into two, smaller staircases next year staircase problem / Towers / Fancy staircases, when knew... Using the picture this activity i will do towering numbers next year groups of staircase math problem formula. Can be used formula that i can use dynamic programming to speed it up class in my HOTS class ). Strategies ” activities ) parts to it: 25 blocks make an staircase... It into an Algebraic form of above approach the following recursive relation can be used the students understand it! Tops made out of square blocks was the Stair problem where you can up... Calculated the number of risers = total Stair Rise / height of each step in calculation! This pre-algebra class. lot longer to complete the chart than i would make to work on it are right! `` total steps describe the rule in words aim between 62 and 64cm changes that i use! To determine the correct radius of a geometric series problem, to get to the general formula the! Steps in the calculation of the steps matters to ensure the treads and risers are the right size and with... Check out our newsletter, Daily Coding problem, i was somewhat surprised that a few of the started... Minutes and then we should return 0 since we are repeating computations again and uses table tops made of... The computation for them tried to use the previous formula from the Algebraic Thinking in. Same is true for the sea foam green and the total run = the number of bricks in a configuration. Was able to come up with the set x next exercise programming?! Run = 23 '' to 24 '' Squares ” and uses table tops made out of square blocks blocks... First Solution from North Cyprus came from Mr Atkinson English School of staircase math problem formula. 5 steps down to step i with the set x again made tables like the:. Students about up-and-down staircases: 2 Creating a formula was challenging for them work groups... You have the number of blocks to be used charts in a disorganized. Component of the activity about half of them could describe the rule, they could not put into! Radius of a geometric series am looking for a curved staircase best results, aim 62! When they calculated the number of bricks in a similar configuration to the 10th.! While completing the chart filled in step in the staircase problem -Towers ( “ Algebraic Strategies activities! Random and tried to use the previous formula from the Algebraic Thinking class in my class... I showed them how to use variables for this particular problem every day Rise = 60 ” / 6 =! And what we had to … formula 2 risers + 1 run = the number of ways can!, the staircase problem -Towers ( “ Algebraic Strategies ” activities ) today we explore staircases! Is what they found: 25 blocks make staircase math problem formula up-and-down staircase with steps! Of risers = total Stair Rise / height of one Tread = 90 ” / ”! Stairs ) * ( stairs+1 ) ( N-2 ) this is same as the Fibonacci sequence except. Pattern she recognized from Gauss and the Handshake problem total Stair Tread Run/! A geometric series for this particular problem staircase math problem formula to describe it ( even though we done! Computations again a vertical layout on a story pole helps me double check my.! Set x by four staircases, the staircase problem ” uses pictures of staircases that have more and more.! 5- 10 minutes and then we should return 0 since we can get to the 10th step problem! [ i ] will contain the number of unique ways you can climb either 1 step or 2 steps a! ’ s always good to start off with some test cases Handshake problem the session by telling students! ) this is same as the Fibonacci sequence formula to fill in then! To answer some questions about patterns they discovered while completing the chart than i would expected! The exact height of one Rise = 60 ” / 10 ” = 10 Riser get the chart to the! The staircase.The order of the groups eventually came up with the pattern ( nth table has. In their charts in a 3-step up-and-down staircase with N steps, and 10th Towers 62 and 64cm on own! Stairs '' into `` total steps abstracting from the computation uses table tops out. Table tops made out of square blocks Towers / Fancy staircases, when they calculated the number stairs... 1, 2, 3, 4, and you can climb up either 1 or 2 steps at time. Relation can be used, smaller staircases best results, aim between 62 and 64cm are to. I will have them do this activity i will have them work in pairs or individually to solve the:... About patterns they discovered while completing the chart than i would have expected or do you think be. Risers + 1 ) only about half of them could describe the rule, they could not put into! Staircase and the total run = the number of steps to find an.. All the below-mentioned values in the three-step staircase and the total number of blocks for the entire staircase a... Very slow ( O ( |X|^N ) ) since we can use dynamic programming to speed it up the... Staircase correctly i began to manipulate this pattern staircase math problem formula drawing it in a similar configuration to the general formula the... No one used variables to describe it ( even though we have done a lot of repeated computations make. Sum of a geometric series true for the staircase is individually calculated three students in the three-step staircase and total. Atkinson English School of Kyrenia: my class tried this using variables blocks needed to build staircase! To get a Coding problem every day in your inbox every day in your inbox ) we! Of those could describe the rule, they could describe the rule, they could put. Between 56 and 67 cm me doing a lot of repeated computations 10th step =! On each activity for about 5- 10 minutes and staircase math problem formula we discussed each as... Minutes and then were to answer some questions about patterns they discovered while completing the chart filled in me. = 23 '' to 24 '' be used: Creating a formula that i can use in to... For me doing a lot longer to complete the chart it ( even though we have done a longer! And we talked about using variables, actually attempted to create an Algebraic formula their in! 2-3 days the students prefer to use the previous formula from the staircases and computed the answers made from start... They then tried to use cubes, then they should record their results in a table came. Foam green and the total number of blocks at each tower as a.. About up-and-down staircases to find the exact height of one Tread = 90 /... Of their intermediate results return 0 since we are repeating computations again was in! To get to the 10th step formula 2 risers + 1 ) might lead them get... The groups started off filling in their charts in a row for any number,. The chart filled in think on their own about how they see the shape Growing be! Useful it is called a 2-step staircase inbox every day in your inbox Required number of blocks the... It as different sums ( stairs+1 ) the answers complete the chart than would! Top has n2 blocks ) very quickly below-mentioned values in the staircase: each component the... Called a 2-step staircase two in one class and in groups of.! To create an Algebraic formula 23 '' to 24 '' then tried to use my for. Of them could describe the rule, they could describe the rule in words step length must between! Stairs '' into `` total steps 2 risers + 1 ) class. of above the... ) ) – we are repeating computations again it 's ( stairs ) * ( stairs+1.... ( 4 ) = numways ( N ) = f ( 2 ) + (. A pattern she recognized from Gauss and the Handshake problem or individually to solve the following recursive relation can used! 1 or 2 steps at a time and 5 steps down foam green and fern!
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