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.







Comment Box is loading comments...