Dividing complex numbers

Solve the first problem 17 problems

Dividing complex numbers ●○○○○ 10 ptsCompute $\dfrac{20+10i}{1-3i}$ . Write the result in the form $a+bi$ and find $a$ and $b$ .
Product of complex numbers divided by i ●○○○○ 10 ptsLet $z_1=1+i$ and $z_2=2-i$ . Write $\frac{z_1z_2}{i}$ in the form $a+bi$ with real numbers $a$ and $b$ . Find $a$ and $b$ .
Inverse of a complex number ●○○○○ 10 ptsFind the real part and the imaginary part of $(7+9i)^{-1}$ . Round each to two decimal places.
Real part of reciprocal on unit circle ●●○○○ 20 ptsLet $z$ be a complex number with $|z| = 1$ and $z \neq -1$ . Show that $\operatorname{Re}\left(\frac{1}{z+1}\right)$ is a constant, and find its value.
Arithmetic-geometric series in complex z ●●○○○ 20 ptsLet $z=1-2i$ . Find the real part and the imaginary part of $\frac{1}{z}+\frac{2}{z^{2}}+\frac{3}{z^{3}}+\cdots$ .
Simplify a complex exponential ratio ●●○○○ 20 ptsLet $\theta$ be an angle such that $\tan \theta = 2$ . Simplify $\frac{e^{2i \theta} - 1}{e^{2i \theta} + 1}$ and write the result in the form $a+bi$ with real numbers $a$ and $b$ . Find $a$ and $b$ .
Term of a complex recurrence ●●●○○ 50 ptsThe sequence of complex numbers $\{z_n\}$ satisfies $z_1=1$ and $z_{n+1}=\overline{z_n}+1+ni$ for $n=1,2,\ldots$ . Write $z_{2015}=a+bi$ , where $a$ and $b$ are real numbers. Find $a$ and $b$ .
Complex sequence with conjugate recurrence ●●●○○ 50 ptsThe complex number sequence $\{z_n\}$ satisfies $z_1=1$ and $z_{n+1}=\overline{z_n}+1+ni$ for $n=1,2,3,\dots$ , where $i$ is the imaginary unit and $\overline{z_n}$ is the conjugate of $z_n$ . Find the real part and the imaginary part of $z_{2015}$ .
Derivative of a complex power ●●●○○ 50 ptsLet $f(z)$ denote the principal branch of $z^{1+i}$ . Find $f'(i)$ . Write $f'(i)=a+bi$ and find $a$ and $b$ . Round each to four decimal places.

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