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Middle Term

 

(i) When n is even

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

i.e. nCn/2 an/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 3rd and 4th both are the middle terms, which are the ((5+1)/2)th and ((5+3)/2)th terms.

Note: 
  * rth 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 = 2nCn (1)2n-n(-x)n = 2nCn(-x)n = (2n)!/n!n!
(-x)n.

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