What is the coefficient of the middle term in the expansion of (1 + 4x + 4x²)⁵?

Concept:

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.

• 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.

• 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.

(1−x)n=∑k=0n(nk)1n−k(−x)k(1−x)n=∑k=0n(nk)1n−k(−x)k(1−x)n=∑k=0n(nk)1n−k(−x)

Calculation:

Given:

(1 + 4x + 4x2)5

⇒ [(1 + 2x)²]⁵

⇒ (1+ 2x)¹⁰ 

Here n = 10 (n is even number)

∴ Middle term =  = 6th term 

Middle Term, T₆ =  T_{5 + 1} = ¹⁰C₅ (1)⁵ (2x)⁵

⇒ × 32x⁵

⇒ 8064 x⁵

∴ The coefficient of the middle term in the expansion of (1 + 4x + 4x2)5  is 8064.