Binomial Theorem for Positive Integral Indices

Introduction

Binomial Theorem for Positive Integral Indices states that “the total number of terms in the expansion is one more than the index”. The nth row of this array gives the coefficients in the expansion of $\mathrm{(a\:+\:b)^{n}}$ in descending powers of a and ascending powers of b; this array is known as the Pascal’s triangle after French mathematician Blaise Pascal (1623-1662).

The triangle is, in fact, much older; it appeared as early as in 1303 in the works of the Chinese mathematician Chu Shin-Chien. Indeed, this was described by Indian mathematician Halayudha in 10th century A.D. as Meru Prastara, 700 years before it was stated by Pascal. Not only that, Halayudha’s Meru Prastara was only a clarification of a rule invented by Pingala more than 1200 years earlier (around 200BC).

It is amazing to know that Pingala, a writer of the third century BC in his Chhandas Sutra gives the following table for finding the number of combinations of n letters taken one, two, three… at a time. First draw a square. Below it and starting from the middle of the lower side, draw two squares on either side. Similarly, draw 3, 4, 5 squares below these, write the number one in the middle of the top square and inside the first and the last square of each row. Inside every other square, the number to be written is the sum of the numbers in the two squares above it and overlapping. Observe that what you have attained is the so-called Pascal triangle.

Indeed, the diagram illustrates that $\mathrm{C_{r}^{n\:+\:1}\:=\:C_{r}^{n}\:+\:C_{r\:-\:1}^{n}}$

In this tutorial, we will learn about the binomial theorem for positive integral indices, we will also learn about the binomial expansions, and middle term of a binomial expansion. We will also learn the formula for the coefficient of a specific term of a binomial expansion and use this knowledge to solve some example questions. Lets begin with the binomial theorem first.

Binomial Theorem for positive integer exponent

Any expression involving only one term like 𝑎, 5, 2𝑥2 i.e. any variable or constant or product of some constants and/or variables with or without some powers is usually called a monomial. If two monomials are connected by ‘+’ or ‘-’ sign, then the resulting expression is called a binomial. For instance, 2+x, 6x-y (i.e., of the form a+x, involving two terms a and x, not both constant at a time) are some binomial expressions.

Binomial Expansion

The subject content of binomial theorem is concerned with such binomial expressions.

So far, you know formulas for expansions of $\mathrm{(a\:+\:b)^{2}}$ and $\mathrm{(a\:+\:b)^{3}}$, which are obtained by directly multiplying (a+b) twice and thrice respectively. But it is not practicable to find (𝑎 + 𝑏)𝑛 by direct multiplication if the index n is a very large positive number. Thus a general formula for the expression of (𝑎 + 𝑏)𝑛 for any positive number n is of great value. Sir Issac Newton innovated this formula in 1664-65. This formula is known as ‘Binomial theorem’ which is stated as follows −

$$\mathrm{(x\:+\:y)^{n}\:=\:c_{0}^{n}x^{n}y^{0}\:+\:c_{1}^{n}x^{n\:-\:1}y^{1}\:+\:c_{2}^{n}x^{n\:-\:2}y^{2}\:+\:.....\:+\:c_{n\:-\:1}^{n}x^{1}y^{n\:-\:1}\:+\:c_{n}^{n}x^{0}y^{n}}$$

For any positive integer n,

Where y, x are real numbers.

Midterms

The binomial expression of $\mathrm{(a\:+\:b)_{n}}$ contains (n+1) terms, so that the number of terms is even or odd according as n is odd or even respectively. If (n+1) is odd, then clearly there is only one middle term, viz., the $\mathrm{(\frac{n}{2}\:+\:1)}$ th term $\mathrm{\frac{tn}{2}\:+\:1}$ in the expansion of $\mathrm{(a\:+\:b)^{n};}$ but if $\mathrm{(n\:+\:1)}$ is even, then obviously there are two middle terms, viz., the $\mathrm{\frac{n\:+\:1}{2}th}$ term and the $\mathrm{(\frac{n\:+\:1}{2}\:+\:1)}$ th term $\mathrm{\frac{tn\:+\:1}{2}\:+\:1}$ in the binomial expansion of $\mathrm{(a\:+\:b)^{n}}$

Coefficient of a term in Binomial Expansion

As we have already seen, binomial expansion of $\mathrm{(a\:+\:b)_{n}}$ contains (n+1) terms. The rth term tr from the beginning and the rth term from the end, say 𝑡` are called the equidistant terms for 𝑟 = 1,2,3, … , 𝑛 + 1. Thus the 1st term 𝑡1 and the last term $\mathrm{t_{n\:+\:1}\:(=\:t)}$ are equidistant terms, the 2nd term 𝑡2 and the last but one term $\mathrm{t_{n}\:(=\:t_{2})}$ are equidistant terms. Now, the rth term $\mathrm{t_{r}}$ from the end is the $\mathrm{[(n\:+\:1)\:-\:r\:+\:1]th}$ i.e., $\mathrm{(n\:+\:2\:-\:r)th}$ term from the beginning.

Hence, the general term of a binomial theorem is

$$\mathrm{t_{r}\:=\:C_{r\:-\:1}^{n}\:a^{n-\:r\:+\:1}x^{r\:-\:1}}$$

Solved Examples

1) Find the term independent of x in $\mathrm{(2x\:+\:\frac{1}{3x^{2}})^{9}}$

Ans. Let the (r+1)th term be independent of x.

Now, $\mathrm{t_{r\:+\:1}\:=\:C_{r}^{9}\:(2x)^{9\:-\:r}(\frac{1}{3x^{2}})^{r}\:=\:C_{r}^{9}\:3^{-r}\:x^{9\:-\:r\:-\:2r}}$

According to the assumption, $\mathrm{x^{9\:-\:r\:-\:2r}\:=\:x^{0};9\:-\:3r\:=\:0;\:r\:=\:3}$

So, the term independent of x is the 4th term.

Now, the required term is $\mathrm{t_{4}\:=\:C_{3}^{9}2^{6}3^{-3}\:=\:\frac{1792}{9}}$

2)If (n+1)th term in the expansion of $\mathrm{(\frac{2}{3}x^{2}\:-\:\frac{1}{3x})^{9}}$ is independent of x, then find the value of n.

Ans. According to the question, the (n+1)th term in the expansion of the given binomial expression is independent of x.

Now,$\mathrm{t_{n\:+\:1}\:=\:C_{n}^{9}(\frac{2}{3}x^{2})^{9\:-\:2}\:(-\frac{1}{3x})^{n}\:=\:C_{n}^{9}(\frac{2}{3})^{9\:-\:n}\:(-\frac{1}{3})^{n}\:x^{18\:-\:3n}}$

Consequently,$\mathrm{x^{18\:-\:3n}\:=\:x^{0};18\:-\:3n\:=\:0;\:n\:=\:6}$

So, the required value of n is 6.

Conclusion

In this tutorial, we learned about the binomial theorem for positive integral indices, we also learned about the binomial expansions, and middle term of a binomial expansion. We also learned the formula for the coefficient of a specific term of a binomial expansion and used this newly gained knowledge to solve some example questions.

The expansion of the expression $\mathrm{(x\:+\:y)_{n}}$ is basically binomial theorem and the expansion goes like

$$\mathrm{(x\:+\:y)^{n}\:=\:c_{0}^{n}x^{n}y^{0}\:+\:c_{1}^{n}x^{n\:-\:1}y^{1}\:+\:c_{2}^{n}x^{n\:-\:2}y^{2}\:+\:.....\:+\:c_{n\:-\:1}^{n}x^{1}y^{n\:-\:1}\:+\:c_{n}^{n}x^{0}y^{n}}$$

It was first proposed by Issac Newton, although this was described by Indian mathematician Halayudha in 10th century A.D. as Meru Prastara. The triangle formed by placing the values in the proper order is known as the Pascal triangle.

FAQs

1. Define what a binomial theorem is?

It is not practicable to find (𝑎 + 𝑏)𝑛 by direct multiplication if the index n is a very large positive number. Sir Issac Newton innovated this formula in 1664-65. This formula is known as ‘Binomial theorem’ which is stated as follows −

$$\mathrm{(x\:+\:y)^{n}\:=\:c_{0}^{n}x^{n}y^{0}\:+\:c_{1}^{n}x^{n\:-\:1}y^{1}\:+\:c_{2}^{n}x^{n\:-\:2}y^{2}\:+\:.....\:+\:c_{n\:-\:1}^{n}x^{1}y^{n\:-\:1}\:+\:c_{n}^{n}x^{0}y^{n}}$$

2. Find the constant term that is present in a binomial theorem?

The constant term is basically the term devoid of any variable. In

$$\mathrm{(a\:+\:y)^{n}\:=\:c_{0}^{n}a^{n}y^{0}\:+\:c_{1}^{n}a^{n\:-\:1}y^{1}\:+\:c_{2}^{n}a^{n\:-\:2}y^{2}\:+\:.....\:+\:c_{n\:-\:1}^{n}a^{1}y^{n\:-\:1}\:+\:c_{n}^{n}a^{0}y^{n}}$$

consider a as the constant, then the first term $\mathrm{c_{2}^{2}a^{n}}$ is the constant

3. State the coefficient present in a binomial theorem.

The general term of a binomial theorem is $\mathrm{t_{r}\:=\:C_{r\:-\:1}^{n}\:a^{n\:-\:r\:+\:1}x^{r\:-\:1}}$. From this value, we can easily say that the coefficient of a general term in a binomial theorem is the general term of a binomial theorem is $\mathrm{c_{r\:-\:1}^{n}\:a^{n\:-\:r\:+\:1}}$ and it is the coefficient of $\mathrm{x^{r\:-\:1}}$.

4. Find the general term that is present in a binomial theorem?

The rth term $\mathrm{t_{t}}$ from the end is the $\mathrm{[(n\:+\:1)\:-\:r\:+\:1]th}$ i.e,, $\mathrm{(n\:+\:2\:-\:r)th}$term from the beginning. Hence, the general term of a binomial theorem is

$$\mathrm{t_{r}\:=\:C_{r\:-\:1}^{n}\:a^{n\:-\:n\:+\:1}\:x^{r\:-\:1}}$$

5. State the number of terms present in a binomial theorem?

If we consider the expansion

$\mathrm{(a\:+\:y)^{n}\:=\:c_{0}^{n}a^{n}y^{0}\:+\:c_{1}^{n}a^{n\:-\:1}y^{1}\:+\:c_{2}^{n}a^{n\:-\:2}y^{2}\:+\:.....\:+\:c_{n\:-\:1}^{n}a^{1}y^{n\:-\:1}\:+\:c_{n}^{n}a^{0}y^{n}}$, then,the number of terms present here is (n+1).