🗊Презентация Binomial theorem

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Binomial theorem, слайд №1Binomial theorem, слайд №2Binomial theorem, слайд №3Binomial theorem, слайд №4Binomial theorem, слайд №5Binomial theorem, слайд №6Binomial theorem, слайд №7Binomial theorem, слайд №8Binomial theorem, слайд №9Binomial theorem, слайд №10
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Binomial theorem
Описание слайда:
Binomial theorem

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Binomial theorem. This is the formula that represents the expression for a positive integer as a polynomial: Note that the sum of the exponents of and is constant and equal to .
Описание слайда:
Binomial theorem. This is the formula that represents the expression for a positive integer as a polynomial: Note that the sum of the exponents of and is constant and equal to .

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- Binomial coefficients, - non-negative integer Example: From the set {1,2,3,4}, select all the possible combinations of two elements, = {1,2} {1,3} {1,4} {2,3} {2,4} {3,4} It turns out six options. Substituting values into the formula, we check the result: binomial coefficient Example
Описание слайда:
- Binomial coefficients, - non-negative integer Example: From the set {1,2,3,4}, select all the possible combinations of two elements, = {1,2} {1,3} {1,4} {2,3} {2,4} {3,4} It turns out six options. Substituting values into the formula, we check the result: binomial coefficient Example

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The properties of binomial coefficients The properties of binomial coefficients 1. The sum of the coefficients of expansion is equal to. It is sufficient to put = 1. Then the right side of Newton's binomial expansion we will have sum of binomial coefficients, and on the left:
Описание слайда:
The properties of binomial coefficients The properties of binomial coefficients 1. The sum of the coefficients of expansion is equal to. It is sufficient to put = 1. Then the right side of Newton's binomial expansion we will have sum of binomial coefficients, and on the left:

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2. The coefficients members equidistant from the ends of the expansion, are equal. 2. The coefficients members equidistant from the ends of the expansion, are equal. This property follows from the relation:
Описание слайда:
2. The coefficients members equidistant from the ends of the expansion, are equal. 2. The coefficients members equidistant from the ends of the expansion, are equal. This property follows from the relation:

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3. The amount of the even terms in the expansion coefficient equal to the sum of the odd terms in the expansion coefficients; each of them is 3. The amount of the even terms in the expansion coefficient equal to the sum of the odd terms in the expansion coefficients; each of them is To prove this we use the binomial: Here the even members are  sign, and the odd -  . As a result turns decomposition 0, therefore, the amount of binomial coefficients are equal to each other, so each of them is: get to prove.
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3. The amount of the even terms in the expansion coefficient equal to the sum of the odd terms in the expansion coefficients; each of them is 3. The amount of the even terms in the expansion coefficient equal to the sum of the odd terms in the expansion coefficients; each of them is To prove this we use the binomial: Here the even members are  sign, and the odd -  . As a result turns decomposition 0, therefore, the amount of binomial coefficients are equal to each other, so each of them is: get to prove.

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Examples of using the binomial of Newton Examples of using the binomial of Newton Example 1. Arrange bean in powers of . We use the binomial theorem: The values of the binomial coefficients consistently find the formula For example
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Examples of using the binomial of Newton Examples of using the binomial of Newton Example 1. Arrange bean in powers of . We use the binomial theorem: The values of the binomial coefficients consistently find the formula For example

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Example 2. Prove formula Example 2. Prove formula Substituting in the formula for expansion value Get the desired result.
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Example 2. Prove formula Example 2. Prove formula Substituting in the formula for expansion value Get the desired result.

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Example 3. Prove ratio Example 3. Prove ratio We use the recurrence relation
Описание слайда:
Example 3. Prove ratio Example 3. Prove ratio We use the recurrence relation

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Thank you for attention
Описание слайда:
Thank you for attention

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