Middle Term

 

(i) When n is even

Middle term of the expansion is the (n/2 + 1)^th term

i.e. ⁿC_n/2 a_n/2 in the expansion of (a + b)n.

(ii) When n is odd

Middle terms of the expansion are the (n/2 + 1)^th term and the ((n+3)/2)^th term.

These are given by,ⁿC_((n-1)/2)  a^((n+1)/2) a^((n-1)/2) and ⁿC_((n+1)/2)  a^((n-1)/2) a^((n+1)/2)  in the expansion of (a + b)n.

e.g. middle term in the expansion of (1+x)⁴ and (1+x)⁵.

Expansion of (1+x)⁴ have 5 terms, so third term is the middle term which is the ((4/2)+1)^th term.

Expansion of (1+x)⁵ have 6 terms, so 3^rd and 4^th both are the middle terms, which are the ((5+1)/2)^th and ((5+3)/2)^th terms.

Note: 
  * r^th term from the end = (n - r + 2)^th term from the beginning. 
  * If there are two middle terms, then the binomial co-efficients of two middle terms will be equal and those two co-efficients will be greatest.

Illustration:

Find the middle term in the expansion of (1 - 2x + x²)ⁿ.

Solution:

We have (1 - 2x + x²)ⁿ = [(1 -x)²]ⁿ = (1 - x)²ⁿ.

Here 2n is an even integer =>((2n/2) + 1)^th i.e. (n+1)^th term will be the middle term.

Now (n+1)^th term in (1-x)²ⁿ = ²ⁿC_n (1)^2n-n(-x)ⁿ = ²ⁿC_n(-x)ⁿ = (2n)!/n!n!
(-x)ⁿ.

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