Introduction,Binomial theorem

Subject: Mathematics

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This note helps you learn what binomial expression mean and how you can expand the binomial expansion in the form of (a+b)^n. Here, n is a positive integer.
Introduction,Binomial theorem

The expression that consists of two terms is known as a binomial expression. Example :

$$\;a\;+\;b,\;x+\frac{1}{y}\;,\;x^2+y\;$$ . When the power of such expressions is a very small positive integer such as 2, 3, 4 it is not difficult to expand them. But if the power is large numbers, let us say ‘n’ then we need a formula a formula for the expansion of the binomial expression. The expansion of the [removed]a+x)n, where the value of n is a positive integer is called the Binomial Theorem. The binomial theorem was first introduced by Sir Issac Newton.

Binomial Theorem :

According to the binomial theorem, for any positive integer n,

$$(a+x)^n\;=\;^nC_0a^nx^0\;+\;^nC_2a^{n-1}x^1\;+\;^nC_3a^{n-2}x^2\;+\;.\;.\;.\;.\;.\;.\;+\;^nC_ra^{n-r}x^r\;+\;.\;.\;.\;.\;.\;+\;^nC_na^{n-n}x^n$$

Proof: We have, $$\;(a+x)\;=\;(a+X)(a+x)(a+x)\;.\;.\;.\;.\;.\;.\;.\;to\;n\;factors$$

In the process of multiplication of n factors, let us start by taking n = 2, 3, and 4. Then the expansions are :$$\;(a+x)^1=a+x$$ $$\;(a+x)^2=a^2+2ab+b^2$$ $$\;(a+x)^3=a^3+3a^2b+3ab^2+b^3$$ $$\;(a+b)^4=a^4+4a^3b+6a^2b^2+4ab^3+b^4$$

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Form the above equation we can see that,

$$\;(a+x)^2=a^2+2ab+b^2\;=^2C_0a^2b^0+^2C_1ab+^2C_2a^0b^2$$

$$\;(a+x)^3=a^3+3a^2b+3ab^2+b^3\;=^3C_0a^3b^0+^3C_1a^2b^1+^3C_2a^1b^2+^3C_3b^3$$

So, the theorem holds good when n= 2, 3.

Let us assume that the theorem is true for all positive integer values of n. Let us say m, is the new value of n, then

$$(a+b)^m\;=\;^mC_0a^mb^0\;+\;^mC_1a^{m-1}b^1\;+\;^mC_2a^{m-2}b^2\;+\;.\;.\;.\;.\;.\;.\;+\;^mC_ra^{m-r}b^r\;+\;.\;.\;.\;.\;.\;+\;^mC_ma^{m-m}b^m$$

Multiplying both sides by (a + b), we get

$$(a+b)^m+1\;=\;(a+b)\;(\;^mC_0a^mb^0\;+\;^mC_1a^{m-1}b^1\;+\;^mC_1a^{m-2}b^2\;+\;.\;.\;.\;.\;.\;.\;+\;^mC_ra^{m-r}b^r\;+\;.\;.\;.\;.\;.\;+\;^mC_ma^{m-m}b^m\;)$$

$$\;=\;^mC_0a^{m+1}b^0\;+\;^mC_1a^{m}b^1\;+\;^mC_2a^{m-1}b^2\;+;.\;.\;.\;.\;.\;+\;^mC_ma^{1}b^m\;+\;.\;.\;.\;.\;.\;.\;^mC_0a^mb^1\;+\;^mC_1a^{m-1}b^2\;+\;^mC_1a^{m-2}b^3\;+\;.\;.\;.\;.\;+\;^mC_mb^{m+1}$$

$$\;=\;a^{m+1}\;+\;(^mC_1\;+\;1)\;a^mx\;+(^mC_2\;+\;^mC_1\;)\;a^{m-1}x^2\;+\;(^mC_3\;+\;^mC_2\;)\;a^{m-2}x^3\;+\;.\;.\;.\;.\;.\;+\;1\;x^{m+1}$$

$$\;=\;^{m+1}C_0a^{m+1}\;+\;^{m+1}C_1a^mx\;+\;^{m+1}C_2a^{m-1}x^2\;+\;.\;.\;.\;.\;.\;.\;+\;^{m+1}C_{m+1}x^{m+1}$$

Therefore when the theorem is true for n = m. The theorem is also true for n = m + 1.

We know that the theorem is true for n = 3, therefore it must be true n = 4

Properties of the expansion of (a + b)n

Some of the important properties on binomial expansion are as follows :

  • The number of terms in the expansion of (a + b)n is n + 1 i.e. one more than the index n.
  • In the successive terms of the expansion, the index of ‘a’ goes on decreasing by unity, starting from n, then n – 1 , and ending with zero, on the contrary, the index of b goes on increasing by unity, starting from 0, then 1, . . . . . . . . . .and ending with n.
  • In any term, the sum of indices of a and b is equal to n.
  • In the expansion of (a + b )n,

The first term T1 = T0 + 1 = nC0 an-0 b0

The second term T2 = T1 + 1 = nC1 an-1 b1

The third term T3 = T2 + 1 = nC2 an-2 b2 . . . . . . . . . . .

Similarly, the (r + 1)th term Tr + 1 = nCr an-r br which is called the general term.

  • The coefficients of the terms equidistant from the beginning and the end are always equal.

Middle term

Now, let us find the middle term or terms of the expansion (a + b)n. We have to consider the cases when n is an even number and when n is an odd number.

  • (i) When n is even

When n is even, we write n = 2m ,where m = 1, 2, 3, . . . . The number of terms after expansion is 2m + 1, which is odd. So, it has only one middle term, namely (m + 1)th term. So,

$$\;t_{m+1}\;=\;^{2m}C_ma^mb^m\;=\;\frac{2m!}{(m!)^2}\;a^m\;b^m$$

$$\therefore\;middle\;term\;t_{m+1}\;=\;t_{\frac{1}{2}n+1}$$

$$\;=\;^nC_{\frac{1}{2}n}a^{n-\frac{n}{2}}x^{\frac{n}{2}}$$

$$\;=\;\frac{n!}{(\frac{1}{2}n!)^2}a^{n-\frac{n}{2}}x^{\frac{n}{2}}$$

  • (ii) When n is odd

When n is odd, we write n = 2m – 1, where m = 1, 2, 3 …., The number of terms after expansion is 2m, which is even. So, there will be two middle terms, namely mth and (m + 1)th term. So,

$$\;t_m\;=\;^{2m-1}C_{m-1}.a^m.x^{m-1}$$

$$\;=\;\frac{(2m-1)!}{m!(m-1)!}\;.a^m.x^{m-1}$$

and,

$$\;t_{m+1}\;=\;^{2m-1}C_m.a^{m-1}.x^m$$

$$\;=\;\frac{(2m-1)!}{m!(m-1)!}\;.a^{m-1}.x^{m}$$

$$\therefore\;the\;middle\;terms\;are$$

$$\;t_m\;=\;t_{\frac{n+1}{2}}\;=\;t_{\frac{n-1}{2}+1}$$

and,

$$t_{m+1}\;=\;t_{\frac{n+1}{2}+1}$$

Taken reference from

( Basic mathematics Grade XII and A foundation of Mathematics Volume II and Wikipedia.com )

Things to remember
  • The expression that consists of two terms is known as a binomial expression.
  • According to the binomial theorem, for any positive integer n,

    $$(a+x)^n\;=\;^nC_0a^nx^0\;+\;^nC_2a^{n-1}x^1\;+\;^nC_3a^{n-2}x^2\;+\;.\;.\;.\;.\;.\;.\;+\;^nC_ra^{n-r}x^r\;+\;.\;.\;.\;.\;.\;+\;^nC_na^{n-n}x^n$$

    • The number of terms in the expansion of (a + b)n is n + 1 i.e. one more than the index n.
    • In the successive terms of the expansion, the index of ‘a’ goes on decreasing by unity, starting from n, then n – 1 , and ending with zero, on the contrary, the index of b goes on increasing by unity, starting from 0, then 1, . . . . . . . . . .and ending with n.
    • In any term, the sum of indices of a and b is equal to n.
    • In the expansion of (a + b )n,
  • It includes every relationship which established among the people.
  • There can be more than one community in a society. Community smaller than society.
  • It is a network of social relationships which cannot see or touched.
  • common interests and common objectives are not necessary for society.

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