The greatest coefficient in the expansion of (1 + x)²ⁿ⁺⁴ is

Concept:

The greatest coefficient will be of the middle term.

General term: 

General term in the expansion of (x + y) n is given by 

Middle terms: The middle terms is the expansion of (x + y) ⁿ depends upon the value of n.

1. If n is even, then the total number of terms in the expansion of (x + y) ⁿ is n+1. So there is only one middle term i.e.  term is the middle term.
2. If n is odd, then the total number of terms in the expansion of (x + y) ⁿ is n+1. So there are two middle terms i.e. and  are two middle terms.

Calculation:

We have to find the greatest coefficient in the expansion of (1 + x)2n+4 

As we know the greatest coefficient will be of the middle term of the expansion.

Here, (2n + 4) is even, there will be only one middle term.

Middle term =  term = (n + 2 + 1)^th term = (n + 3)^th term

General term in the expansion of (x + y)³ is given by 

∴ T_{(n + 3)} = T_{(n + 2)+1} = ²ⁿ⁺⁴C_n+2 xⁿ⁺²