Question

Find the middle term in the expansion of `(x^2+1)^18`?

Answer 1

Middle term in the expansion of `(x^2+1)^18` is `48620x^18`

Explanation:

In the expansion of `(x+a)^n`, there are `(n+1)` terms and `r^(th)` term is `C_r^nx^(n-r)a^r`, where `r` ranges from `0` to `18`.

Hence there are `18+1=19` terms in the expansion of `(x^2+1)^18`

and middle term is `(19+1)/2=10` i.e. `10^(th)` term and as terms start from `r=0`, `10^(th)` term is for `r=9`

and it is `C_9^18(x^2)^(18-9)*1^9`

= `(18*17*16*15*14*13*12*11*10)/(1*2*3*4*5*6*7*8*9)*x^18*1`

= `48620x^18`