Middle Term in Binomial Expansion

Binomial expression

Before knowing about binomial expansion and its middle term, let us know what a binomial is. An algebraic expression with two terms is called a binomial. The terms of binomial expressions are connected through a plus or minus sign.

Binomial expansion or binomial theorem

When a binomial term is raised to a non-zero positive exponent (except 1), the binomial is expanded. But when the exponents are more significant, it’s a tedious process to find the solution manually. Here, when the binomial expansion formulas come into the picture. 

Formula of binomial theorem:

Let n ∈ N,x,y,∈ R then

(x + y)n=r=nr=0ΣnCrxn-r . yr  where,

nCr=n!/(n-r)!r!

Important points regarding binomial expansion 

• The total number of terms in the expansion of (a+b)n is (n+1)

• The sum of exponents of a and b is always n.

• nC0, nC1, nC2, … .., nCn are binomial coefficients and can also be written as  C0, C1, C2, ….., Cn.

• The equidistant  binomial coefficients  from the beginning and from the ending are equal; nC0 = nCn, nC1 = nCn-1 , nC2 = nCn-2 ,….. etc.

• Binomial coefficients can also be found using  Pascal’s Triangle.

Terms in the Binomial Expansion

There are middle and general terms to simplify binomial expansion. 

Other than general and middle term,  there are other terms too, including

• Independent Term

• Determining a Particular Term

• Numerically greatest term

• The ratio of Consecutive Terms/Coefficients

General Term of a Binomial Expansion

The binomial expansion formula is given below:

If (x + y)n= nC0xn+ nC1 xn-1. y + nC2xn-2 . y2 + … + nCnyn

Then, the General Term = Tr+1 = nCr xn-r. yr

• The General Term in (1 + x)n is nCrxr

• In the binomial expansion of (x + y)n, the rth term from the end is (n – r + 2)th    

Example: Find the number of terms in (4 + 4x +x2)50

Answer: 

(4 + 4x + x2)50 = [(2 + x)2]50= (2 + x)100

The number of terms = (100 + 1) = 101

Middle Term in Binomial Expansion

To understand the middle term of a binomial expansion, we consider binomial expressions of the form (x+y)n 

Now, If (x+y)n  = nCr.xn-r.yr, it contains (n + 1) terms, with the middle term depending on n.

For the Middle Term of a Binomial Expansion, there are two types of instances as per the value of n. These include even and odd values of n.

First Case with even n:

If n is an even number, then (n + 1) is an odd number. To get the middle term, use the expansion formula.

So, the middle term would be the [(n+1+1)/2]th term which is the [(n/2) + 1]th term.

Let us consider an example. In the expansion of (x+3)4, we find the middle term as n+1= 4+1 = 5. So, the middle term would be  [(n/2) + 1]th term or [(4/2 + 1)] term= 3rd term

Thus, the third term would be the middle term in the expansion of (x+3)4.

Second Case with odd n:

If n is an odd number, then (n+1) would be even. Now, in the case of an even number of terms in the expansion, there would be two middle terms instead of one. The middle terms would be [(n+1)/2]th term and {[(n+1)/2] + 1}th term.

Considering an example of odd n terms, we use the expression (x+3)5. For the middle term, we first find the (n+1)th term. So, (n+1) = (5 + 1)= 6 which is even. This means that there would be two middle terms in the expansion of (x+3)5.

Now, the two middle terms would be the [(n+1)/2]th term and {[(n+1)/2] + 1}th term. So, the  [(n+1)/2]th term = [(5+1)/2] = [6/2]= 3rd term and

{[(n+1)/2] + 1}th term = {[(5+1)/2] + 1} = {[(6)/2] + 1} = [3 + 1] = 4th term.

So, the middle terms in the expansion of (x+3)5 would be the third and the fourth terms.

Independent Term

The middle term in binomial expansion also includes a case called the independent term. In this case, we use the following expression as:

(a + 1/a)2n where a is not zero. Here, the middle term would be given by [(n/2) + 1]th term as 2n is even. So,

Substituting the value of n in the middle term formula, we get :

[(n/2) + 1] = [(2n/2) + 1] = [n + 1]th term.

The value of the middle term would be 2nCn an(1/a)n= 2nCn

Since the value of the middle term is independent of the value of a, this term is known as an independent term.

Properties of Binomial Coefficients

Binomial coefficients are the integers of each variable term in a binomial expression. 

Here are the important properties of binomial coefficients that will help solve the mathematical problems easily. 

• C0 + C1 + C2+ … + Cn = 2n

• C0+ C2+ C4 + … = C1 + C3 + C5 + … = 2 n-1

• C0– C1 + C2 – C3+ … +(−1)n . nCn = 0

• nC1 + 2.nC2+ 3.nC3 + … + n.nCn = n.2n-1

• C1− 2C2+ 3C3 − 4C4 + … +(−1)n-1 Cn = 0 for n > 1

• C02 + C12 + C23 + …Cn2= [(2n)!/ (n!)2]=2nCn

Conclusion

Binomial Theorem gives the expansion of a binomial for every positive integral n. The expansion coefficients are organised in an array. Pascal’s triangle is the name given to this array. According to the magnitude of n, there are two sorts of occurrences for the Middle Term of a Binomial Expansion. These may be even or odd values of n. In the case of an even value of n, the middle term would be the [(n/2)+1]th term. In the case of an odd value of n, there would be two middle terms. These middle terms would be the [(n+1)/2]th term and {[(n+1)/2] + 1}th term.