Discrete math Mathematics Binomial formula and an extended number 3x + 2x 2x+1 SUPERVISORS PHD TRAN NGUYEN AN DISCRETE MATH MEMBERS LƯU TÙNG THU TRÀ BÙI HIỀN group 6 HỒNG NGỌC INTRODUCTION Mathematics has a particularly important position in the subjects in high schools, it is the basis of many other subjects It is a subject that many students love because of its abstract thinking so that they can freely discover new things when they go to learn it The application of the binomial formula has the.
DISCRETE MATH Binomial formula and an extended 3x + 2x number SUPERVISORS: PHD TRAN NGUYEN AN 2x+1 MEMBERS LƯU TÙNG C Ọ G N G N Ồ H group THU TRÀ BÙI HIỀN INTRODUCTION Mathematics has a particularly important position in the subjects in high schools, it is the basis of many other subjects It is a subject that many students love because of its abstract thinking so that they can freely discover new things when they go to learn it The application of the binomial formula has the effect of reviewing and systematizing knowledge and affirming the practicality of the content of knowledge If students practice solving this form of math, it not only helps students master the mathematical knowledge system, but also contributes to training math problem solving skills, skills to apply math knowledge to practice, and developing mathematical thinking study for students With that in mind, my group's essay presents the topic: ''The binomial formula and an extended number'' 1 COMBINATION SYMBOL t n e i c ffi e o Binomial c The symbol binomial coefficient is the coefficient of in the binomial expansion Combinatorial formula In Mathematics, combinatorics is a way of selecting elements from a larger group regardless of the order In smaller cases the number of combinations can be counted For example for three fruits, an apple, an orange and a pear, there are three ways to combine the two fruits from this set: an apple and a pear; an apple and an orange; a pear and an orange By definition, the concatenation of n elements is a subset of the parent set S containing elements, the subset of k distinct elements belonging to S and unordered The number of convolutional combinations of n elements is equal to the binomial coefficient We have the formula PASCAL'S TRIANGLE AND THE FORMATION OF NEWTON'S BINOMIAL FORMULA l a i m o n i b e h t f o n o i t a m r o The f formula - Special cases of the binomial theorem have been known since at least the 4th century BC, when the Greek mathematician Euclid mentioned a special case of the binomial theorem for the exponent of - Al-Karaji described the triangular model of the binomial coefficients and gave proofs for the binomial theorem and Pascal's triangle by mathematical induction - In 1544, Michael Stifel introduced the term "binomial coefficients" and showed how to use them to represent through using "Pascal's triangle" - Newton is sometimes considered the founder of mathematical analysis 2.2.NEWTON’S BINOMIAL STORY -Newton Isaac found the following binomial expansion formula, which is called Newton's binomial - In Europe, the arithmetic triangle was first found in the work of the German mathematician Stiffel M Published in 1544 -One hundred years later, completely independent of each other, English mathematicians Borigon (1624), French mathematician Fermat (1636) and French mathematician Pascal (1654) came up with the perfect formula about the coefficients of Newton's binomial Pascal Triangle - In mathematics, Pascal's triangle is a triangular array of binomial coefficients -The rows of Pascal's triangle are listed by convention starting with the top row n=0 (row 0) The entries in each row are numbered from the left end with k=0 and are often staggered relative to the numbers in adjacent rows -In row (top row), there is a unique Each number of each subsequent row is constructed by adding the number above and to the left with the number above and to the right, treating empty entries as n=0 n=1 n=2 n=3 n=4 n=5 1 10 10 Prove by induction on variable n The formula = -With n=0, we have =1= Assume the formula is correct in the case of 0N, with N, now we will go to prove it correctly in case n=N+1 Realyy, we have: = = = = - With case k N we have = + According to the inductive assumption, the formula holds for the case n, so that == = Thence inferred = PROVE NEWTON’S BINOMIAL FORMULA n le ab ri va on on ti uc d in by e ov Pr With n=0, n=1 it is obvious that we have soething to prove Assume the formula is correct in the case of n N, with N, now we will go to prove it correctly in case n=N+1 We have Notice that according to Pascal's triangle construction formula, we have: So we have proved the correct formula for the case n=N+1 3.SOME BASIC PROPERTIES Recalling Newton's binomial Session Theorem: With a, b being real numbers and n being positive integers, we have In the expression on the right hand side of formula (1) we have a) Number of terms n+1 b) The number of terms with the exponent of a decreasing from n to 0, the exponent of b increasing from to n, but the sum of the exponents of a and b in each term is always n c) The coefficients of each term equidistant from the first and last two terms are equal Consequences 1.With a=b=1, we have With a = 1; b = -1, we have 0= BASIC FORMULAS RELATING TO NEWTON BINOMIAL DEVELOPMENT ALSO WE HAVE SOME OTHER RECIPES AS FOLLOWING In addition, from the formula k we can expand the following formula Signs of problems using Newton's binomial in proof of equality problems - When it is necessary to prove an equality or inequality where and i are consecutive natural numbers - If the problem is for expansion then the coefficient of is such that the equation a (n−i)+bi=m has a solution i∈N - In the expression then we multiply both sides by and then take the derivative - In the expression then we choose the appropriate value of x=a 4.SOME EXTENSIONS OF THE BINOMIAL FORMULA 1,Polynomial formula ).() = + = 3, n 4, + + ) = MATHEMATICAL FORMS RELATED TO NEWTON'S BINOMIAL CONSISTS OF • The problem of binomial expansion and the proof of fundamental equality • • • The problem of the largest coefficient Prove the equality Application of derivatives in the proof of combinatorial equality • • Application of integrals in the proof of combinatorial equality Application of complex numbers to prove combinatorial equality SELF PRACTICE EXERCISES CONCLUSION In the current university's advanced math program, discrete modules play an important role and are distributed right from the first semester In it, the binomial formula and some of its extensions play a dominant role It helps to solve many problems in many fields Therefore, binomial formulas and some extensions have many wide applications in life In the process of studying the essay due to limited time, after completing the essay, we still continue to study more deeply about the binomial formula and some extensions We are looking forward to receiving suggestions from teachers to improve our essay Thank You! See you in our next presentation ... from this set: an apple and a pear; an apple and an orange; a pear and an orange By definition, the concatenation of n elements is a subset of the parent set S containing elements, the subset of... to S and unordered The number of convolutional combinations of n elements is equal to the binomial coefficient We have the formula PASCAL'S TRIANGLE AND THE FORMATION OF NEWTON'S BINOMIAL FORMULA. .. binomial theorem for the exponent of - Al-Karaji described the triangular model of the binomial coefficients and gave proofs for the binomial theorem and Pascal's triangle by mathematical induction