Question

In the expansion of ${(x^2+1+\frac{1}{x^2})}^n$, $n\in N$,

• A. number of terms is $2n+1$
• B. coefficient of constant term is $2^{n-1}$
• C. coefficient of ${\left(x\right)}^{2n-2}$ is $n$
• D. coefficient of $x^2$ in $n$

Answer 1

Answer: (A), (C)

coefficient of ${\left(x\right)}^{2n-2}$ is $n$
number of terms is $2n+1$

${(x^2+1+\frac{1}{x^2})}^n{=}^n{C}_0{+}^n{C}_1(x^2+\frac{1}{x^2}){+}^n{C}_2{(x^2+\frac{1}{x^2})}^2+...{+}^n{C}_n{(x^2+\frac{1}{x^2})}^n$
This contain each of the term
$x^0,x^2,x^4,...x^{2n},x^{-2},x^{-4}...x^{-2n}$
Coefficient of constant term
${=}^n{C_0+}^n{C}_2\left(2\right){+}^n{C}_4\left({}^4{C}_2\right)+\left({}^n{C}_6\right)\left({}^6{C}_3\right)+...\ne 2^{n-1}$
Coefficient of $x^{2n-2}$ in ${}^n{C}_{n-1}=n$
Coefficient of $x^2$ is ${{}^{n}c}_1+\left({}^n{C}_3\right)\left({}^3{C}_1\right)+\left({}^n{C}_5\right)\left({}^5{C}_1\right)+...>n$