Suppose $n$ is a natural number such that $|i + 2i^2 + 3i^3 + \ldots + ni^n| = 18\sqrt{2}$,where $i = \sqrt{-1}$. Then,$n$ is

Let $f(x) = x^4 + ax^3 + bx^2 + c$ be a polynomial with real coefficients such that $f(1) = -9$. Suppose that $i\sqrt{3}$ is a root of the equation $4x^3 + 3ax^2 + 2bx = 0$,where $i = \sqrt{-1}$. If $\alpha_1, \alpha_2, \alpha_3$,and $\alpha_4$ are all the roots of the equation $f(x) = 0$,then $|\alpha_1|^2 + |\alpha_2|^2 + |\alpha_3|^2 + |\alpha_4|^2$ is equal to:

If $x = -2 + \sqrt{-3}$,then the value of $2x^4 + 5x^3 + 7x^2 - x + 38$ is equal to

If $\alpha$ is a root of the equation $x^2+x+1=0$ and $\sum_{k=1}^n\left(\alpha^k+\frac{1}{\alpha^k}\right)^2=20$,then $n$ is equal to

If ${z_1} = 1 + 2i$ and ${z_2} = 3 + 5i$, then $\operatorname{Re} \left( \frac{{\bar{z}_2}{z_1}}{{z_2}} \right)$ is equal to

Let $z = 1 + ai$ be a complex number,$a > 0$,such that $z^3$ is a real number. Then the sum $1 + z + z^2 + .... + z^{11}$ is equal to