Middle Terms for Positive Integral Index

The number of terms in the expansion of (a + b)ⁿ depends on the index n. The index n is either even or odd.

Case 1: n is even.

Let n = 2k

The number of terms is n+1 i.e., 2k+1.

The middle term has k terms before it.

Case 2: n is odd.

Let n = 2k+1

The number of terms is n+1 i.e., (2k + 1) + 1 = 2k + 2.

In this case, there are two middle terms and are after k terms.

Thus, in (a + b)ⁿ:

Note:

The middle terms may be easily found out by using the following method:

i) When n is even, we add the even number 2 to n and divide by 2 to get the middle term i.e.,  term.

ii) When n is odd, we add the odd numbers 1 and 3 to n and divide by 2 to get the middle terms i.e.,  terms.

Example:

Find the middle terms in the expansion of:

Suggested answer: