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Didactical material for lesson:“Newton’s binomial ”

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Newton’s binomial

Lesson aim:

•Formula presentation (a+b)n, a,bє Ł si nєAt*•Finding the properties for the terms coefficients from this binominal development.•Applications

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5)( ba and 5)( ba and 5)( ba and 5)( ba and 5)( ba Answer:

and 5)( ba 4)( ba Write the formula for : (a+b)³, (a+b)², find a method to calculate :

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What can you say about

letter’s coefficients?

What can you say about the

number of terms from each

development?

What can you say about

letter’s exponents?

Answer the following

questions:

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Answers:

•The coefficients of the extreme terms and those equally distant from the extreme terms are equal. •The exponents of its power are decreasing from the most high to 0.•The exponents of “b” power increases from 0 to the highest.•The highest exponent for “a” and for “b” is the exponent at which the binomial rises.•The number of terms from the development exudes with 1 the exponent at which the binomial it is risen.

•The coefficients of the extreme terms and those equally distant from the extreme terms are equal. •The exponents of its power are decreasing from the most high to 0.•The exponents of “b” power increases from 0 to the highest.•The highest exponent for “a” and for “b” is the exponent at which the binomial rises.•The number of terms from the development exudes with 1 the exponent at which the binomial it is risen.

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Calculate the numbers in the given situations: a) n=1; b) n=2; c) n=3; d) n=4; e) n=5.

Answer: Using the combination formula

and using the complementary combination formula

we obtain:

Calculate the numbers in the given situations: a) n=1; b) n=2; c) n=3; d) n=4; e) n=5.

Answer: Using the combination formula

and using the complementary combination formula

we obtain:

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We observe the following situation in the second problem:

a) n1; b) n2; c) n3; d) n4; e) n5, namely:

a)

b)

1 1

1 2 1

1 3 3 1

1 4 6 4 1

1 5 10 10 5

c)

d)

e) 1

The coefficients from the development are exactly the numbers obtained calculating in the situations given:

Thus grouped it can be seen a way to calculate these numbers from closely to closely.( Pascal’s triangle)

a) n1; b) n2; c) n3; d) n4; e) n5, namely:

a)

b)

1 1

1 2 1

1 3 3 1

1 4 6 4 1

1 5 10 10 5

c)

d)

e) 1

The coefficients from the development are exactly the numbers obtained calculating in the situations given:

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Newton’s formula

There is: Binomial’s theorem , a,bє Ł si nєAt*, then

known also as Newton’s formula.Isaac Newton, English mathematician, astronomer, physician (1643-1727)

Demonstration using mathematic induction method:Step I. Verification : P(1): ……. Independent work …..

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Theorem demonstration :

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Specifications regarding Newton’s formula:

1.the coefficients are called binomial coefficients of the development and are in number of n+1.

Is necessary to make a distinction between the binomial coefficient of a term and the numerical coefficient of the same term.

2. Those n+1 are

3. The natural numbers are called binomial coefficients of odd rank, and the numbers are called binomial coefficients of even rank.

4. In Newton’s formula the exponents of a powers are decreasing from n to 0, and exponents of b power are increasing from 0 to n.

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5. The binomial coefficients of the extreme terms and those equally distant from the extreme terms are equal :

6. If the power exponent is even, n=2k, then the development has 2k+1 terms, and the middle term has the highest binominal coefficient :

If the power exponent is odd, n=2k+1, then the development has 2k+2 terms and there are two terms in the middle of the development with equally binomial coefficients and of highest value

7. An important role, in resolving problems related with Newton’s binomial, is played by the general term having the rank k+1:

Specifications regarding Newton’s formula ( continuation)

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1. Calculate using Newton’s binomial. After developing the binomial using the formula complete the

following:a) = ………………………b) The binomial coefficient of is …………………..c) The coefficient of is ………………………d) The free term of the development is ………………………….e) The term that contains is …………………….f) The term that contains is …………………….

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Aplication:

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Thus:

a)

b) The binomial coefficient of is

c) The coefficient of is

d) The free term

e) The term that contain is

f) there is no term that contains

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Answer:

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Aplication:

Application: 2. Calculate using Newton’s formula and answer the following questions:a) = …………………..b) the binomial coefficient of is …………….?c) the coefficient of is …….?d) Re (z)=……………….e) Im (z)=……………….

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In conclusion :

a)

b) the binomial coefficient of is

c) The coefficient of is

d)

e)

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Answer:

In conclusion :

a)

b) the binomial coefficient of is

c) The coefficient of is

d)

e)

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Aplication:

3. It is given the binomial You have to determine:a) the third term of the development.b) the middle term.c) the rank of the term that contains d) the term that contains e) the free term in the development (do not develop the binomial!)

3. It is given the binomial You have to determine:a) the third term of the development.b) the middle term.c) the rank of the term that contains d) the term that contains e) the free term in the development (do not develop the binomial!)

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Answer:

3. The general term is: a) we take k=2 and we obtain

b) how n=8 means that the development has 9 terms and the middle term is

c) in order to find the term that contains we use from its formula the factor x with its exponent:

d) we repeat the reasoning and we finde) Analogue

there is no free term

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Answer:

is biggest of the development terms if

where n≥10

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Application:

4. To determine knowing that the tenth term of the binomial development is biggest of the development terms.4. To determine knowing that the tenth term of the binomial development is biggest of the development terms.

4. To determine knowing that the tenth term of the binomial development is biggest of the development terms.

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Aplication:

5. Being given the binomiala) determine the number of terms from the developmentb) find out how many rational terms does the development has.c) how many irrational terms does the development has?

5. Being given the binomiala) determine the number of terms from the developmentb) find out how many rational terms does the development has.c) how many irrational terms does the development has?

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Answer:

5. a)The binomial has in development a number of 101 terms b) the general term formula is

or else it is written :

there are 17 rational termsc) in conclusion there are 101-17=84 irrational terms

5. a)The binomial has in development a number of 101 terms b) the general term formula is

or else it is written :

there are 17 rational termsc) in conclusion there are 101-17=84 irrational terms

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Identities in the combination calculus

Using the Newton’s formula for binomial development

There can be deduced some interesting identities in which binomial coefficients intervene.

• Particularised in Newton’s formula a=b=1 we find :

the sum of the development of the binomial coefficients is 2ⁿ

• In the same formula taking a=1 and b=-1 we obtain:

the alternating sum of the binomial coefficients is 0

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Or :

the sum of the binomial coefficients of odd rank is

Subtracting the two sum we obtain

or

The sum of the binomial coefficients of even rank is

Adding the two sums member by member we obtain:

Or :

the sum of the binomial coefficients of odd rank is

Subtracting the two sum we obtain

or

The sum of the binomial coefficients of even rank is

Identities in the combination calculus( continuation)

Adding the two sums member by member we obtain: Adding the two sums member by member we obtain: Adding the two sums member by member we obtain: Adding the two sums member by member we obtain:

Or :

the sum of the binomial coefficients of odd rank is

Subtracting the two sum we obtain

or

The sum of the binomial coefficients of even rank is

Adding the two sums member by member we obtain:

Or :

the sum of the binomial coefficients of odd rank is

Subtracting the two sum we obtain

or

The sum of the binomial coefficients of even rank is

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Aplication:

6. Calculate the sum :

a) using the equality

for n,k є Ł and n ≥ k

b) using the complementary combination’s formula

for n,k є Ł and n ≥ k

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Answer:

a) demonstration of the formula

Thus the sum is rewritten

a) demonstration of the formula

Thus the sum is rewritten

a) demonstration of the formula

Thus the sum is rewritten

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b) we rewrite the sum using the complementary combination formula and we obtain :

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Answer (continuation):

We add the two sums :

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Aplication:

7. Demonstrate the equality

for n,k є Ł and n ≥ k

and then calculate the sum

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Answer:

7. The demonstration of the formula :

With this formula we rewrite each term of the sum

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Test

It is given the binomial :1. How many terms does the development has?2.Which is the rank of the middle term?3. Which is the sum of the binomial coefficients of thisbinomial? Using the general term formula, find out :4.The rank of the term that contains x².5. How many rational terms does the development has?

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1. Find out the binomial term of the development which contains if the sum of the binomial coefficients is 128.

2. It is considered the development a) determine “n” so that the sum of the first three coefficients of the

development be 97.b) for n=8 check out if there exists a term that contains c) for n=80 find out the sum of the coefficients of the development.

3) a) write the complex number z=1+i under trigonometric form and then calculate with Moivre’s formula.

b) Develop using Newton’s formulac) equalizing the equalities from a) and b) let the equalities be deducted:

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Homework: