Question

(i) The sum of the binomial coefficients of the expansion $(x+\frac{1}{x}{)}^n$ is $2^n$.
(ii) The term independent of $x$ in the expansion of ${(x+\frac{1}{x})}^n$ is $0$ when is even.
Which of the above statements is correct?

• A. Only (i)
• B. Only (ii)
• C. both (i) and (ii)
• D. neither (i) nor (ii)

Answer 1

The correct option is A Only (i)

Sum of all the coefficients in a binomial expansion of type $(a{x}^p+b{x}^q)$ for $a>0$ $b>0$ is always $2^n$.
Now if $n$ is even there will be $n+1$ terms. Hence, there will odd number of terms.
Hence, the middle term will be the term independent of in $(x+\frac{1}{x}{)}^n$.
Hence, $T_{(n/2+1)}$ is the term independent of $x$.
Hence only (i) is true.