# hockey stick pattern in pascal's triangle

Computers and access to the internet will be needed for this exercise. 1 k ∑k=1n∑j=1kj=(n+23)=(n+2)!(n−1)!(3)!=n(n+1)(n+2)6. Then change the direction in the diagonal for the last number. , This method can be continued indefinitely to develop an identity for the sum of any power of natural numbers. − , and noticing that 1 {\displaystyle n+1} \end{aligned}6n(n+1)(n+2)​21​k=1∑n​k2k=1∑n​k2​=21​k=1∑n​k2+21​(2n(n+1)​)=6n(n+1)(n+2)​−4n(n+1)​=6n(n+1)(2n+1)​.​. 3 Sign up, Existing user? 3. are not on the committee. . {\displaystyle 0\leqslant i\leqslant n} 2 ⩽ {\displaystyle k\in \mathbb {N} ,k\geqslant r} ⩽ Examine the numbers in each "hockey stick" pattern within Pascal's triangle. . The sum of the first nnn triangular numbers can be expressed as. x The Fibonacci sequence is related to Pascal's triangle in that the sum of the diagonals of Pascal's triangle are equal to the corresponding Fibonacci sequence term. For whole numbers nnn and r (n≥r),r\ (n \ge r),r (n≥r), ∑k=rn(kr)=(n+1r+1). The oranges are arranged such that there is 1 top orange; the second top layer has 2 more oranges than the top; the third has 3 more oranges than the second, and so on. 1 1 … The pattern is similar to the shape of a "hockey-stick". = n Circling these elements creates a "hockey stick" shape: 1+3+6+10=20. You just need the row number and the length of the hockey stick. Combinatorics in Pascal’s Triangle Pascal’s Formula, The Hockey Stick, The Binomial Formula, Sums. + Starting from any of the 1s on the outermost edge, ... (hence the “hockey-stick” pattern). ∈ 1 For example, 3 is … The Pascal Triangle. □\sum\limits_{k=1}^{n}{k}=\binom{n+1}{2}=\frac{(n+1)!}{(n-1)!(2)! Count the rows in Pascal’s triangle starting from 0. The pattern known as Pascal’s Triangle is constructed by starting with the number one at the “top” or the triangle, and then building rows below. {\displaystyle n-k+1} disjoint cases. The curve starts at a low-activity level on the X-axis for a short period of time. For the past decade, the king of Mathlandia has forced his subjects to build a pyramid in his honor. ∑k=1nk=∑k=1n(k1).\sum\limits_{k=1}^{n}{k}=\sum\limits_{k=1}^{n}\binom{k}{1}.k=1∑n​k=k=1∑n​(1k​). Pascal’s triangle was originally developed by the Chinese Blaise Pascal was the first to actually realize the importance of it and it was named after him Pascal's triangle is a math triangle that can be used for many things. Print each row with each value separated by a single space. angle both have hockey stick type patterns, patterns of sums within shapes, and have the natural numbers, triangular numbers, tetrahedral numbers, and pentatope numbers hidden within. The king would tolerate no waste, so he ordered one of his subjects to be sacrificed for each leftover slab of stone. I thought this was a great genre for students that love hands on projects, and visual aides. Pascal'’ triangle… n The big hockey stick theorem is a special case of a general theorem which our goal is to introduce it. Now, the 15 lies on the Hockey Stick line (the line of numbers in this case in the second column). {\displaystyle x} It’s lots of good exercise for students to practice their arithmetic. : is known as the hockey-stick[1] or Christmas stocking identity. i It is intended for about 4th grade level, so it doesn't go through all possible patterns found in Pascal's triangle, but just some simple ones: the sums of the rows, counting numbers in a diagonal, and triangular numbers. □​. Another famous pattern, Pascal’s triangle, is easy to construct and explore on spreadsheets. Hockey-stick theorem. − Hockey Stick Identity. , Substituting these identities, the identity for the sum of squares of the first nnn positive integers can be developed: n(n+1)(n+2)6=12∑k=1nk2+12(n(n+1)2)12∑k=1nk2=n(n+1)(n+2)6−n(n+1)4∑k=1nk2=n(n+1)(2n+1)6.\begin{aligned} 3 If you add up all the numbers in any row, what do you get? . to Somehow, Pascal gained the credit for the triangle. □\begin{aligned} That last number is the sum of every other number in the diagonal. The hockey stick chart formation illustrates that urgent action may be required to understand a phenomenon or find a solution for the drastic shift in data points. Now consider the sum of the sum of squares of positive integers: ∑k=1n∑j=1kk2=∑k=1nk(k+1)(2k+1)6=13∑k=1nk3+12∑k=1nk2+16∑k=1nk.\begin{aligned} thing I visualized was the triangle. \frac{n(n+1)^2(n+2)}{12} The beauty of Pascal’s Triangle is that it’s so simple, yet so mathematically rich. ⩽ Ask the students if they see any patterns. . Determine the sum of the terms in each row of Pascal's triangle. Hockey Stick Identity Start at any of the " 1 1 " elements on the left or right side of Pascal's triangle. {\displaystyle n} 1 As you can see from the figure 1+3+6=10 shown in red and similarly for green hockey stick pattern 1+7+28+84=120. (See the picture for an example of a pyramid 3 levels high constructed in the same way). Then, each subsequent row is formed by starting with one, and then adding the two numbers directly above. \ _\squarek=r∑n​(rk​)=(r+1n+1​). This can also be expressed with binomial coefficients: ∑k=1nk3=6(n+34)−6(n+23)+(n+12). 1 ∑k=rn+1(kr)=(n+1r+1)+(n+1r)=(n+1)!(n−r)!(r+1)!+(n+1)!(n−r+1)!r!=(n−r+1)(n+1)!(n−r+1)!(r+1)!+(r+1)(n+1)!(n−r+1)!(r+1)!=(n+2)!(n−r+1)!(r+1)!=(n+2r+1). j As this sum can be expressed as the sum of binomial coefficients, it can be computed with the hockey stick identity: The sum of the first nnn positive integers is, ∑k=1nk=∑k=1n(k1)=(n+12). people. ; Inductive step , We can divide this into In Pascal's triangle, the sum of the elements in a diagonal line starting with 1 1 is equal to the next element down diagonally in the opposite direction. Hockey Stick Pattern: We can even make a hockey stick pattern in Pascal’s triangle. Showing the Substituting these variables in the identity above gives the hockey stick identity: ∑k=rn(kr)=(n+1r+1). Skip to 5:34 if you just want to see the relationship. This can be done in (j+q−2q−2)\displaystyle\binom{j+q-2}{q-2}(q−2j+q−2​) ways for each value of j.j.j. 1 These values are the binomial coefficients. … For . Start at any 1 and proceed down the diagonal ending at any number. {\displaystyle n-k+1} The hockey stick pattern is one of many found in Pascal's triangle. \end{aligned}k=1∑n​j=1∑k​k2​=k=1∑n​[2(3k+2​)−(2k+1​)]=2(4n+3​)−(3n+2​)=12n(n+1)(n+2)(n+3)​−6n(n+1)(n+2)​=12n(n+1)2(n+2)​.​, n(n+1)2(n+2)12=13∑k=1nk3+n(n+1)(2n+1)12+n(n+1)12=13∑k=1nk3+2n(n+1)212∑k=1nk3=n2(n+1)24.\begin{aligned} The sum of the first nnn triangular numbers was found previously using the hockey stick identity: ∑k=1nk(k+1)2=n(n+1)(n+2)6.\sum\limits_{k=1}^{n}\frac{k(k+1)}{2}=\frac{n(n+1)(n+2)}{6}.k=1∑n​2k(k+1)​=6n(n+1)(n+2)​. This leads to the more well-known formula for triangular numbers. The king decreed the pyramid to be constructed with cubic stone slabs. The brilliance behind this work is magnificent! Figure 2: The Hockey Stick The “hockey-stick rule”: Begin from any 1 on the right edge of the triangle and follow the numbers left and down for any number of steps. Although other mathematicians in Persia and China had independently discovered the triangle in the eleventh century, most of the properties and applications of the triangle were discovered by Pascal. n This sum can be alternatively computed using binomial coefficients and the hockey stick identity: ∑k=1n∑j=1kk2=∑k=1n[2(k+23)−(k+12)]=2(n+34)−(n+23)=n(n+1)(n+2)(n+3)12−n(n+1)(n+2)6=n(n+1)2(n+2)12.\begin{aligned} , {\displaystyle 1\leqslant x\leqslant n-k+1} ′ + Following are the first 6 rows of Pascal’s Triangle. That’s why it has fascinated mathematicians across the world, for hundreds of years. − . 2. \sum\limits_{k=1}^{n}\sum\limits_{j=1}^{k}{k^2} &=\sum\limits_{k=1}^{n}\left[2\binom{k+2}{3}-\binom{k+1}{2}\right] \\ \\ Now we can sum the values of these : We can form a committee of size □​​, Combinatorial Proof using Identical Objects into Distinct Bins. Change it into a sum of the two above! k + &= 2\binom{n+3}{4}-\binom{n+2}{3} \\ \\ The value at the row and column of the triangle is equal to where indexing starts from . }=\frac{n(n+1)(n+2)}{6}.\ _\squarek=1∑n​j=1∑k​j=(3n+2​)=(n−1)!(3)!(n+2)!​=6n(n+1)(n+2)​. It can be represented as. J. Garvin|Looking For Patterns In Pascal's Triangle Slide 16/19 pascal's triangle and applications Patterns in Pascal's Triangle Another interesting pattern in Pascal's Triangle is often called \hockey stick" pattern. And in this link you can read about MANY more patterns in Pascal's Triangle -- such as magic 11's, square numbers, Fibonacci's sequence, and the "hockey stick pattern." A very unique property of Pascal’s triangle is – “At any point along the diagonal, the sum of values starting from the border, equals to the value in … Consider the previous identity for the sum of squares of positive integers: ∑k=1nk2=n(n+1)(2n+1)6=2(n+23)−(n+12).\sum\limits_{k=1}^{n}{k^2}=\frac{n(n+1)(2n+1)}{6}=2\binom{n+2}{3}-\binom{n+1}{2}.k=1∑n​k2=6n(n+1)(2n+1)​=2(3n+2​)−(2n+1​). So, there are 210\color{#D61F06}{210}210 ways to select 3 balls from the same row. The "Hockey Stick" property and the less well-known Parallelogram property are two characteristics of Pascal's Triangle that are both intruiging but relatively easy to prove. One of the most interesting Number Patterns is Pascal's Triangle (named after Blaise Pascal, a famous French Mathematician and Philosopher). The hockey stick identity is often used in counting problems in which the same amount of objects is selected from different-sized groups. In a business, the chart may be representative of, for example, large problems within the sales processSales and Collection CycleTh… Computers and access to the internet will be needed for this exercise. If you start with row 2 and start with 1, the diagonal contains the triangular numbers. □​. b) Does your pat… Imagine that there are mmm identical objects to be distributed into qqq distinct bins such that some bins can be left empty. By a direct application of the stars and bars method, there are, ways to do this. In pairs investigate these patterns. Each of these elements corresponds to the binomial coefficient (n1),\binom{n}{1},(1n​), where nnn is the row of Pascal's triangle. This pattern is like Fibonacci’s in that both are the addition of two cells, but Pascal’s is spatially different and produces extraordinary results. 2 For this, we don't need to create the complete triangle. \end{aligned}12n(n+1)2(n+2)​k=1∑n​k3​=31​k=1∑n​k3+12n(n+1)(2n+1)​+12n(n+1)​=31​k=1∑n​k3+122n(n+1)2​=4n2(n+1)2​.​. □​. It is useful when a problem requires you to count the number of ways to select the same number of objects from different-sized groups. \sum_{k=1}^{n}{k^2}&=\frac{n(n+1)(2n+1)}{6}. Pascal’s triangle was originally developed by the Chinese Blaise Pascal was the first to actually realize the importance of it and it was named after him Pascal's triangle is a math triangle that can be used for many things. some secrets are yet unknown and are about to find. The sum of the numbers inside the stick will equal the number that is below the last number and not in the same diagonal. 1 Feb 18, 2013 - Explore the NCETM Primary Magazine - Issue 17. Catalan numbers in Pascal's Triangle appear as an algebraic combination of four neighbors in two rows Hockey Stick Pattern ... Next, I was thinking of all the patterns in Pascal’s Triangle. This can then be computed with the hockey stick identity: ∑k=39(k3)=(104)=210.\sum\limits_{k=3}^{9}\binom{k}{3} = \binom{10}{4} = 210.k=3∑9​(3k​)=(410​)=210. people in, ways. The exercise could be structured as follows: Groups … The sum of the squares of the first nnn positive integers is, ∑k=1nk2=n(n+1)(2n+1)6=2(n+23)−(n+12). There are many wonderful patterns in Pascal's triangle and some of them are described above. Practice their arithmetic can get the hockey stick the trinomial Triangle too little modification ) to Pascal 's is. Me… Hockey-stick addition n+1n+1​ ) ​=1=1.​ stone slabs fascinating pattern is one of found! That adds the two above the last number the tip of these hockey sticks a class there is identity. 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