General Term and Middle Term

For the most part, the binomial theorem is useful in determining the expanded value of an algebraic expression of the type (x + y)n. Finding the value of (x + y)2, (x + y)3, (a + b + c)2 is straightforward and may be accomplished by algebraically multiplying the exponent value by the number of times it appears in the equation. However, calculating the expanded form of (x + y)17 or other similar expressions with larger exponential values necessitates a significant amount of computation. Using the binomial theorem, it is possible to make things a little easier.

 When applying this binomial theorem expansion, the exponent value might be either a negative number or a fraction. 

 The algebraic expansion of powers of a binomial is described by the binomial theorem or binomial expansion. In this theorem, the polynomial “(a + b)n” can be expanded into a sum of terms of the form “axzyc,” where the exponents z and c are non-negative integers and z + c = n, and the coefficient of each term is a positive integer that depends on the values of n and B.

General term of binomial expansion:

The General Term of Binomial Expansion of (x + y)n is as follows, 

            Tr+1 = nCr. xn-r . yr

• In the binomial expansion, the General Term is represented by Tr+1

• It is necessary to utilise the General term expansion in order to find the words indicated in the preceding formula

• The supplied expansion needs to be expanded in order to locate the terms in the binomial expansion

• The binomial expansion of the equation (a + b)n will be as follows:

 (a+b)n = nC0. an + nC1. an-1. b + nC2. an-2. b2 + …. + nCn. bn

 It is T1 = nC0.an that is the first term in the sequence

• The second term in the series is T2 = nC1.an-1.b, and it is the second term in the series

• The third term in the series is T3 = nC2.an-2.b2

• The nth term in the series is Tn= nCn.bn .The series has a total of n terms

Middle term of the binomial expansion:

If (x + y)n = nCr.xn-r.yr has (n + 1) terms, with the middle term depending on the value of n.

 For the Middle Term of a Binomial Expansion, we have two possible scenarios:

If n is even:

If n is an even integer, we convert it to an odd number and consider (n + 1) to be odd, with (n/2 + 1) serving as the middle component in the equation. Simply said, if n is an odd number, we regard it to be an even number. 

If n is an even number, then (n + 1) is an odd number. To find out the middle word, do the following: 

Take, for example, the common phrase for binomial expansion, which is

          Tn/2+1 = nCn/2.xn-n/2. yn/2

 Now, in the foregoing equation, we replace “r” with “n/2” to obtain the middle term . 

• Tr+1 = Tn/2 + 1 

• Tn/2+1 = nCn/2.xn-n/2.yn/2

If n is odd:

Assuming that n is an odd number, we convert it to an even number and consider (n + 1) to be even, with (n + 1/2) and (n + 3/2) as the middle terms between (n + 1/2) and (n + 3/2). For the most part, we consider odd numbers to be even when they aren’t.

If n is an odd number, we have two middle terms. To locate the middle term, use the following formula: 

Take, for example, the common phrase for binomial expansion, which is 

         T(n-1) /2 = nC(n-1) /2. xn-(n-1) /2. y(n-1) /2

 Or, 

          T(n+1) /2 = nC(n+1) /2. xn-(n+1) /2. y(n+1) /2

 In this scenario, we substitute “r” with the two alternative values that were previously mentioned.

• When we compare one term to (n + 1/2) terms, we obtain (r + 1) terms.

                     r + 1 = n + 1/2 

                    r = n + 1/2 -1 

                    r = n -½

• When we compare (r + 1) with (n + 3/2), we get the second middle term.

                   r +1 = n +3/2

                    r = n + 3/2 – 1

                    r = n + ½

 The two middle terms when n is odd are (n – 1/2) and (n + 1/2).

Conclusion:

The Binomial Expansion is in the middle of its term. According to what we know, the expansion of (a + b)n has an even number of terms (n + 1). We can write the middle term or terms of (a + b)n using the value of n as a starting point. For the most part, the binomial theorem is useful in determining the expanded value of an algebraic expression of the type (x + y)n. Finding the value of (x + y)2, (x + y)3, (a + b + c)2 is straightforward and may be accomplished by algebraically multiplying the exponent value by the number of times it appears in the equation. 

The algebraic expansion of powers of a binomial is described by the binomial theorem or binomial expansion. In this theorem, the polynomial “(a + b)n” can be expanded into a sum of terms of the form “axzyc,” where the exponents z and c are non-negative integers and z + c = n, and the coefficient of each term is a positive integer that depends on the values of n and B.

                    Tr+1 = nCr. xn-r . yr

In the binomial expansion, the General Term is represented by Tr+1.