The binomial formula and an extended number

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.

Binomial formula and an extended

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.

The symbol binomial coefficient is the coefficient of

in the binomial expansion.

Binomial coefficient

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

2 PASCAL'S TRIANGLE AND THE

FORMATION OF NEWTON'S BINOMIAL

FORMULA.

The formation of th e binomial

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 2.

- 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

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 0 (top row), there is a unique 1 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 0

n=0 1 n=1 1 1 n=2 1 2 1 n=3 1 3 3 1 n=4 1 4 6 4 1 n=5 1 5 10 10 5 1

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: = = 1 =

=

- With case 1 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.

- With n=0, n=1 it is obvious that we have soething to prove Assume the formula is correct in the case of 0 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

Prove by induction on variable n

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 là n+1.

b) The number of terms with the exponent of a decreasing from n to 0, the exponent of b increasing from 0 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.

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

- 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.

Signs of problems using Newton's binomial in proof of equality problems.

4.SOME EXTENSIONS OF THE BINOMIAL

FORMULA

5 MATHEMATICAL FORMS RELATED TO

NEWTON'S BINOMIAL.

• The problem of binomial expansion and the proof of

fundamental equality.

equality.

• Application of integrals in the proof of combinatorial equality.

equality

CONSISTS OF

6 SELF PRACTICE EXERCISES

7 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.