Question

Prove that if n is even then in expansion of (1 + x)ⁿ, coefficient of middle term will be ((1.3.5....(n - 1))/(2.4.6 ....n))2ⁿ

If n is odd, then coefficient of both the middle term will be ((1.3.5....n)/(2.4.6....(n + 1)))2ⁿ

Answer 1

If n is even in (1 + x)  term we take (1 + x)²ⁿ in place of (1 + x)ⁿ then ‘n’ = 2n is also even

Now, middle term = (2n/2 + 1)^th term

= (n + 1)^th

Similarly, if n is odd, then we take (1 + x)²ⁿ⁺¹ in place of (1 + x)ⁿ, then ‘n’ = 2n + 1 is also odd. 

Now middle term are