Let us now find the Middle Terms in the binomial expansion: (x + y)n
Below are listed 2 cases.
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.