The binomial expansion of ( x k +(1/ x 2 k ))3 n , n ∈ N contains a term independent of x

Question

The binomial expansion of ( x k +(1/ x 2 k ))3 n , n ∈ N contains a term independent of x (A) only if k is an integer (B) only if k is a natural number (C) only if k is rational (D) for any real k.

Tardigrade Admin · Accepted Answer

T r +1= 3 n C r x 3 nk -3 rk ⇒ D where r = n

Q. The binomial expansion of $(x^{k} + \frac{1}{x ^{2 k}})^{3 n}, n \in N$ contains a term independent of $x$

only if $k$ is an integer

only if $k$ is a natural number

only if $k$ is rational

for any real $k$ .

$T_{r + 1} =^{3 n} C_{r} x^{3 nk - 3 r k} \Rightarrow D$ where $r = n$

Q. The binomial expansion of (xk+x2k1​)3n,n∈N contains a term independent of x

only if k is an integer

only if k is a natural number

only if k is rational

for any real k.