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Some Results on Binomial Coefficients and Negative or Fractional Indices | Mathematics (Maths) Class 11 - Commerce PDF Download

In this explainer, we will learn how to use the binomial expansion to expand binomials with negative and fractional exponents.
Recall that the binomial theorem tells us that for any expression of the form ( 𝑎 + 𝑏 𝑥 ) 𝑛 where 𝑛 is a natural number, we have the expansion
Some Results on Binomial Coefficients and Negative or Fractional Indices | Mathematics (Maths) Class 11 - Commerce

In particular, we can take 𝑎 = 𝑏 = 1 .

Theorem: Generalized Binomial Theorem, 𝑎 = 𝑏 = 1 Case

The expansion
Some Results on Binomial Coefficients and Negative or Fractional Indices | Mathematics (Maths) Class 11 - Commerce

is valid when 𝑛 is negative or a fraction (or even an irrational number). In this case, the binomial expansion of ( 1 + 𝑥 ) 𝑛 (where 𝑛 is not a positive whole number) does not terminate; it is an infinite sum. Furthermore, the expansion is only valid for | 𝑥 | < 1.

Since the expansion of ( 1 + 𝑥 ) 𝑛 where 𝑛 is not a positive whole number is an infinite sum, we can take the first few terms of the expansion to get an approximation for ( 1 + 𝑥 ) 𝑛 when | 𝑥 | < 1 . Let us look at an example where we calculate the first few terms.

Example 1: Finding Terms of a Binomial Expansion with a Negative Exponent


Write down the first four terms of the binomial expansion ofSome Results on Binomial Coefficients and Negative or Fractional Indices | Mathematics (Maths) Class 11 - Commerce
Answer Recall that the generalized binomial theorem tells us that for any expression of the form ( 1 + 𝑥 ) 𝑛 where 𝑛 is a real number, we have the expansion
Some Results on Binomial Coefficients and Negative or Fractional Indices | Mathematics (Maths) Class 11 - Commerce

When 𝑛 is not a positive integer, this is an infinite series, valid when | 𝑥 | < 1
Note that we can rewrite Some Results on Binomial Coefficients and Negative or Fractional Indices | Mathematics (Maths) Class 11 - Commerceas ( 1 + 𝑥 ) -1 .This is an expression of the form ( 1 + 𝑥 ) 𝑛 ,with 𝑛 = − 1 .Therefore, the generalized binomial theorem tells us that
Some Results on Binomial Coefficients and Negative or Fractional Indices | Mathematics (Maths) Class 11 - Commerce=1−𝑥+𝑥2−𝑥3+⋯,
 which is an infinite series, valid when | 𝑥 | < 1 .The first four terms of the expansion are 1 − 𝑥 + 𝑥2 − 𝑥3.
We can also use the binomial theorem to expand expressions of the form ( 1 + 𝑏 𝑥 ) 𝑛 for a constant 𝑏 .

Theorem: Generalized Binomial Theorem, 𝑎 = 1 Case


The expansion
(1+𝑏𝑥)n
Some Results on Binomial Coefficients and Negative or Fractional Indices | Mathematics (Maths) Class 11 - Commerce
is an infinite series when 𝑛 is not a positive integer. It is valid whenSome Results on Binomial Coefficients and Negative or Fractional Indices | Mathematics (Maths) Class 11 - Commerce

Example 2: Finding a Missing Value in a Binomial Expansion and Finding the Coefficient of a Term in the Expansion


In the binomial expansion of ( 1 + 𝑏 𝑥 ) -5 ,the coefficient of 𝑥 is − 1 5 .Find the value of the constant 𝑏 and the coefficient of x2 .
Ans:
Recall that the generalized binomial theorem tells us that for any expression of the form ( 1 + 𝑏 𝑥 ) 𝑛 where 𝑛 is a real number, we have the expansion
(1+𝑏𝑥)n
Some Results on Binomial Coefficients and Negative or Fractional Indices | Mathematics (Maths) Class 11 - Commerce

When 𝑛 is not a positive integer, this is an infinite series, valid when | 𝑏 𝑥 | < 1 or | 𝑥 | <Some Results on Binomial Coefficients and Negative or Fractional Indices | Mathematics (Maths) Class 11 - Commerce

We begin by writing out the binomial expansion of ( 1 + 𝑏 𝑥 ) -5 up to and including the term in x2 :
Some Results on Binomial Coefficients and Negative or Fractional Indices | Mathematics (Maths) Class 11 - Commerce

We are told that the coefficient of 𝑥 here is equal to − 1 5 ;that is,
− 5 𝑏 = − 1 5
𝑏 = 3 . To find the coefficient of x2 ,we can substitute the value of 𝑏 back into the expansion to get
To find the coefficient of x2 ,we can substitute the value of 𝑏 back into the expansion to get
Some Results on Binomial Coefficients and Negative or Fractional Indices | Mathematics (Maths) Class 11 - Commerce

Therefore, the coefficient of x2 is 135 and the value of the constant 𝑏 is 3.
More generally still, we may encounter expressions of the form ( 𝑎 + 𝑏 𝑥 ) 𝑛 .Such expressions can be expanded using the binomial theorem. However, the theorem requires that the constant term inside the parentheses (in this case, 𝑎 ) is equal to 1. So, before applying the binomial theorem, we need to take a factor of 𝑎 𝑛 out of the expression as shown below:
Some Results on Binomial Coefficients and Negative or Fractional Indices | Mathematics (Maths) Class 11 - Commerce

We now have the generalized binomial theorem in full generality. 

Theorem: Generalized Binomial Theorem

The expansion
Some Results on Binomial Coefficients and Negative or Fractional Indices | Mathematics (Maths) Class 11 - Commerce
Some Results on Binomial Coefficients and Negative or Fractional Indices | Mathematics (Maths) Class 11 - Commerce

where 𝑛 is not a positive integer is an infinite series, valid whenSome Results on Binomial Coefficients and Negative or Fractional Indices | Mathematics (Maths) Class 11 - Commerce

Let us look at an example of this in practice.

Example 3: Finding Terms of a Binomial Expansion with a Negative Exponent and Stating the Range of Valid Values


Write down the first four terms of the binomial expansion ofSome Results on Binomial Coefficients and Negative or Fractional Indices | Mathematics (Maths) Class 11 - Commercestating the range of values of 𝑥 for which the expansion is valid.
Ans:
Recall that the generalized binomial theorem tells us that for any expression of the form ( 𝑎 + 𝑏 𝑥 ) 𝑛 where 𝑛 is a real number, we have the expansion
Some Results on Binomial Coefficients and Negative or Fractional Indices | Mathematics (Maths) Class 11 - Commerce
Some Results on Binomial Coefficients and Negative or Fractional Indices | Mathematics (Maths) Class 11 - Commerce
When 𝑛 is not a positive integer, this is an infinite series, valid whenSome Results on Binomial Coefficients and Negative or Fractional Indices | Mathematics (Maths) Class 11 - Commerce
Applying this to Some Results on Binomial Coefficients and Negative or Fractional Indices | Mathematics (Maths) Class 11 - Commercewe have 
Some Results on Binomial Coefficients and Negative or Fractional Indices | Mathematics (Maths) Class 11 - Commerce

Some Results on Binomial Coefficients and Negative or Fractional Indices | Mathematics (Maths) Class 11 - Commerce

We can now expand the contents of the parentheses:
Some Results on Binomial Coefficients and Negative or Fractional Indices | Mathematics (Maths) Class 11 - Commerce
Some Results on Binomial Coefficients and Negative or Fractional Indices | Mathematics (Maths) Class 11 - Commerce
Therefore, the first four terms of the binomial expansion ofSome Results on Binomial Coefficients and Negative or Fractional Indices | Mathematics (Maths) Class 11 - Commerceare
Some Results on Binomial Coefficients and Negative or Fractional Indices | Mathematics (Maths) Class 11 - Commerce
The expansion is valid forSome Results on Binomial Coefficients and Negative or Fractional Indices | Mathematics (Maths) Class 11 - Commerce

A classic application of the binomial theorem is the approximation of roots. Suppose we want to find an approximation of some root √ 𝑝 .The idea is to write down an expression of the form ( 𝑐 + 𝑑 𝑥 )   that we can approximate for some small 𝑥 (generally, smaller values of 𝑥 lead to better approximations) using the binomial expansion. The value of 𝑥 should be of the formSome Results on Binomial Coefficients and Negative or Fractional Indices | Mathematics (Maths) Class 11 - Commerce ,where 𝑆 is a perfect square and √ 𝑆 = 𝑠 ( 𝑆 = 1 0 0 or 𝑆 = 4 0 0 are often good choices). Then, we haveSome Results on Binomial Coefficients and Negative or Fractional Indices | Mathematics (Maths) Class 11 - Commerce
Some Results on Binomial Coefficients and Negative or Fractional Indices | Mathematics (Maths) Class 11 - Commerce

The trick is to choose 𝑐 and 𝑑 so that 𝑐 𝑆 + 𝑑 = 𝑅 𝑝 where 𝑅 is a perfect square, so √ 𝑅 = 𝑟 for some positive integer 𝑟 .Then, we have
Some Results on Binomial Coefficients and Negative or Fractional Indices | Mathematics (Maths) Class 11 - Commerce
Some Results on Binomial Coefficients and Negative or Fractional Indices | Mathematics (Maths) Class 11 - Commerce
If our approximation using the binomial expansion gives us the valueSome Results on Binomial Coefficients and Negative or Fractional Indices | Mathematics (Maths) Class 11 - Commerce ,then we can recover an approximation for √ 𝑝 as follows:
Some Results on Binomial Coefficients and Negative or Fractional Indices | Mathematics (Maths) Class 11 - Commerce

Let us see how this works in a concrete example. 

Example 4: Finding an Approximation Using a Binomial Expansion


By finding the first four terms in the binomial expansion of Some Results on Binomial Coefficients and Negative or Fractional Indices | Mathematics (Maths) Class 11 - Commerce and then substituting in 𝑥 = 0 . 0 1 ,find a decimal approximation for √3 .
Ans: The goal here is to find an approximation for √ 3 .Note that the numbers Some Results on Binomial Coefficients and Negative or Fractional Indices | Mathematics (Maths) Class 11 - Commerce together with the 1 and 8 in √ 1 + 8 𝑥 have been carefully chosen. Indeed, substituting in the given value of 𝑥 ,we get
Some Results on Binomial Coefficients and Negative or Fractional Indices | Mathematics (Maths) Class 11 - Commerce
Some Results on Binomial Coefficients and Negative or Fractional Indices | Mathematics (Maths) Class 11 - Commerce

Thus, if we use the binomial theorem to calculate an approximation 𝑉 toSome Results on Binomial Coefficients and Negative or Fractional Indices | Mathematics (Maths) Class 11 - Commerceat the value 𝑥 = 0 . 0 1 ,then we will get an approximation to √ 3 because
Some Results on Binomial Coefficients and Negative or Fractional Indices | Mathematics (Maths) Class 11 - Commerce

which implies
Some Results on Binomial Coefficients and Negative or Fractional Indices | Mathematics (Maths) Class 11 - Commerce
So, let us write down the first four terms in the binomial expansion of Some Results on Binomial Coefficients and Negative or Fractional Indices | Mathematics (Maths) Class 11 - Commerce
Recall that the generalized binomial theorem tells us that for any expression of the form ( 1 + 𝑏 𝑥 ) 𝑛 where 𝑛 is a real number, we have the expansion

(1+𝑏𝑥) 𝑛
Some Results on Binomial Coefficients and Negative or Fractional Indices | Mathematics (Maths) Class 11 - Commerce

When 𝑛 is not a positive integer, this is an infinite series, valid whenSome Results on Binomial Coefficients and Negative or Fractional Indices | Mathematics (Maths) Class 11 - Commerce

In this example, we have
Some Results on Binomial Coefficients and Negative or Fractional Indices | Mathematics (Maths) Class 11 - Commerce
=1+4𝑥−8𝑥+32𝑥+⋯.
We can now evaluate the sum of these first four terms at 𝑥 = 0 . 0 1 :
𝑉=1+4×0.01−8×(0.01 )+32 × (0.01 ) = 1+0.04−0.0008+0.000032=1.039232.
Therefore, we have
Some Results on Binomial Coefficients and Negative or Fractional Indices | Mathematics (Maths) Class 11 - Commerce

Some Results on Binomial Coefficients and Negative or Fractional Indices | Mathematics (Maths) Class 11 - Commerce
Comparing this approximation with the value appearing on the calculator for √3=1.732050807 … ,we see that this is accurate to 5 decimal places.
When making an approximation like the one in the previous example, we can calculate the percentage error between our approximation and the true value. The absolute error is simply the absolute value of difference of the two quantities: |−| true value approximation .To find the percentage error, we divide this quantity by the true value, and multiply by 100. We can calculate the percentage error in our previous example:
We can calculate the percentage error in our previous example:
Some Results on Binomial Coefficients and Negative or Fractional Indices | Mathematics (Maths) Class 11 - Commerce
Some Results on Binomial Coefficients and Negative or Fractional Indices | Mathematics (Maths) Class 11 - Commerce

We can also use the binomial theorem to approximate roots of decimals, particularly in cases when the decimal in question differs from a whole number by a small value 𝑥 ,as in the next example.

Example 5: Using a Binomial Expansion to Approximate a Value

Write down the binomial expansion of Some Results on Binomial Coefficients and Negative or Fractional Indices | Mathematics (Maths) Class 11 - Commerce in ascending powers of 𝑥 up to and including the term in x2 and use it to find an approximation for Some Results on Binomial Coefficients and Negative or Fractional Indices | Mathematics (Maths) Class 11 - Commerce .Give your answer to 3 decimal places.
Ans: 
We want to approximate Some Results on Binomial Coefficients and Negative or Fractional Indices | Mathematics (Maths) Class 11 - CommerceWe notice that 26.3 differs from 27 by 0.7 = 7×0.1 .Therefore, if we evaluate Some Results on Binomial Coefficients and Negative or Fractional Indices | Mathematics (Maths) Class 11 - Commerce at 𝑥 = 0 . 1 ,then we will get Some Results on Binomial Coefficients and Negative or Fractional Indices | Mathematics (Maths) Class 11 - Commerce.We are going to use the binomial theorem to approximate Some Results on Binomial Coefficients and Negative or Fractional Indices | Mathematics (Maths) Class 11 - Commerce .Recall that the generalized binomial theorem tells us that for any expression of the form ( 𝑎 + 𝑏 𝑥 ) 𝑛 where 𝑛 is a real number, we have the expansionSome Results on Binomial Coefficients and Negative or Fractional Indices | Mathematics (Maths) Class 11 - Commerce
Some Results on Binomial Coefficients and Negative or Fractional Indices | Mathematics (Maths) Class 11 - Commerce

When 𝑛 is not a positive integer, this is an infinite series, valid whenSome Results on Binomial Coefficients and Negative or Fractional Indices | Mathematics (Maths) Class 11 - Commerce

Let us write down the first three terms of the binomial expansion of Some Results on Binomial Coefficients and Negative or Fractional Indices | Mathematics (Maths) Class 11 - Commerce
Some Results on Binomial Coefficients and Negative or Fractional Indices | Mathematics (Maths) Class 11 - Commerce
Some Results on Binomial Coefficients and Negative or Fractional Indices | Mathematics (Maths) Class 11 - Commerce

Evaluating the sum of these three terms at 𝑥 = 0 . 1 will give us an approximation for Some Results on Binomial Coefficients and Negative or Fractional Indices | Mathematics (Maths) Class 11 - Commerce as follows:Some Results on Binomial Coefficients and Negative or Fractional Indices | Mathematics (Maths) Class 11 - Commerce
and
Some Results on Binomial Coefficients and Negative or Fractional Indices | Mathematics (Maths) Class 11 - Commerce

=2.97385002286….
Comparing this approximation with the value appearing on the calculator forSome Results on Binomial Coefficients and Negative or Fractional Indices | Mathematics (Maths) Class 11 - Commercewe see that it is accurate to four decimal places. Rounding to 3 decimal places, we have Some Results on Binomial Coefficients and Negative or Fractional Indices | Mathematics (Maths) Class 11 - Commerce

Some Results on Binomial Coefficients and Negative or Fractional Indices | Mathematics (Maths) Class 11 - Commerce

The document Some Results on Binomial Coefficients and Negative or Fractional Indices | Mathematics (Maths) Class 11 - Commerce is a part of the Commerce Course Mathematics (Maths) Class 11.
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FAQs on Some Results on Binomial Coefficients and Negative or Fractional Indices - Mathematics (Maths) Class 11 - Commerce

1. What are binomial coefficients?
Ans. Binomial coefficients are numbers that appear in the expansion of a binomial raised to a positive integer power. They represent the coefficients of the terms in the expansion and are calculated using the formula nCr = n! / (r!(n-r)!), where n and r are non-negative integers.
2. Are binomial coefficients only defined for positive integer indices?
Ans. No, binomial coefficients can also be defined for negative or fractional indices. In such cases, the formula for calculating the coefficients is modified using the concept of the gamma function. The gamma function is an extension of the factorial function to real and complex numbers.
3. How can binomial coefficients be calculated for negative or fractional indices?
Ans. To calculate binomial coefficients for negative or fractional indices, the formula nCr = Γ(n+1) / (Γ(r+1)Γ(n-r+1)) is used, where Γ is the gamma function. The gamma function can be evaluated using various mathematical techniques, such as series expansions or numerical approximation methods.
4. What are some properties of binomial coefficients with negative or fractional indices?
Ans. Binomial coefficients with negative or fractional indices have several interesting properties. One property is that they can be expressed in terms of binomial coefficients with positive indices using the reflection formula. Another property is that they satisfy certain recurrence relations, such as Pascal's identity, which relates binomial coefficients with consecutive indices.
5. Can binomial coefficients with negative or fractional indices be interpreted geometrically?
Ans. Yes, binomial coefficients with negative or fractional indices can have geometric interpretations. For example, when the index is a negative integer, the binomial coefficient can represent the number of subspaces of a given dimension in a vector space. When the index is a fractional number, the binomial coefficient can represent the number of paths in a lattice grid from one point to another.
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