Middle Term in Binomial Theorem


6. Middle Term in the Binomial Theorem:

Let us now find the Middle Terms in our binomial expansion: (x + y)n

Two cases arise depending on index n.

Case 1: when n is odd:

There are two middle terms in Binomial Theorem.

One is T( n + 1)/2 and the other is T( n + 3)/2

For eg,

if the binomial index n is 5, an odd number, then the two middle terms are:

T( n + 1)/2 = T( 5 + 1)/2 = T3 &
T( n + 3)/2 = T( 5 + 3) / 2 = T4

The above method is somewhat round about. Let us see a more simple and straight way of finding middle term in Binomial Theorem.

If index n is 5, then number of terms is 6.

So, the two middle terms are (6/2)th term i.e., 3rd term which is T3 And the immediately next term namely (6/2)th+1 i.e., 4th term which is T4

Note: Here, you should note that 6 stands for number of terms in the binomial expansion, and not the index n. Index n is 5 here.

Case 2: when n is even

When the binomial index n is even, we can see there is only one middle term. And it is T( n/2 + 1)

For example,
if index n = 6, an even number, then the middle term will be

T( n/2 + 1) = T( 6/2 + 1) = T4

And you know how to find this middle term with the method of number of terms in the expansion. I leave that method to you to avoid confusion by discussing two methods. It is always better to stick to one method. With practice, that one method becomes a second nature and occurs to the mind lightning fast. With two methods in mind, the mind gets indecisive as to use which one of the two.