Middle Term of Binomial Expansion

If (x + y)ⁿ =  ⁿC_r.x^{n – r.}y^r  , it has (n + 1) terms and the middle term will depend upon the value of n.

We have two cases for the Middle Term of a Binomial Expansion:

If n is Even 

If n is the even number then we make it into an odd number and consider (n + 1) as odd and (n/2 + 1) as the middle term. In simple, if n is even then we consider it as odd.

Suppose n is the even so, (n + 1) is odd. To find out the middle term:

Consider the general term of binomial expansion i.e. 

• Now we replace “r ” with “n/2” in the above equation to find the middle term
• T_r+1 = T_n/2 + 1
• T_{n/2 + 1} = ⁿC_n/2.x^{n – n/2}.y^n/2

Binomial theorem or expansion describes the algebraic expansion of powers of a binomial. According to this theorem, it is possible to expand the polynomial “(a + b)^n“ into a sum involving terms of the form “ax^zy^c“, the exponents z and c are non-negative integers where z + c = n, and the coefficient of each term is a positive integer depending on the values of n and b. 

Example: If n = 4

(a + b)⁴ = a⁴ + 4a³y + 6a²b² + 4ab³ + b