27.5: Review of complex numbers, sequences, and the binomial theorem
- Page ID
- 68486
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Multiply and write the answer in standard form: \[(-4)(\cos(207^\circ)+i\sin(207^\circ))\cdot 2(\cos(33^\circ)+i\sin(33^\circ)) \nonumber \]
- Answer
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\(4+i 4 \sqrt{3}\)
Divide and write the answer in standard form: \[\dfrac{3(\cos(\frac{\pi}{3})+i\sin(\frac{\pi}{3}))}{15(\cos(\frac{\pi}{2})+i\sin(\frac{\pi}{2}))} \nonumber \]
- Answer
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\(\dfrac{\sqrt{3}}{10}-i \dfrac{1}{10}\)
Find the magnitude and directional angle of the vector \[\vec{v}=\langle -7, -7\sqrt{3}\rangle \nonumber\]
- Answer
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magnitude \(\|\langle-7,-7 \sqrt{3}\rangle\|=14\), directional angle \(\theta=\frac{4 \pi}{3}=240^{\circ}\)
Determine if the sequence is an arithmetic or geometric sequence or neither. If it is one of these, then find the general formula for the \(n\)th term \(a_n\) of the sequence.
- \(54, -18, 6, -2, \dfrac 2 3, \dots\)
- \(2, 4, 8, 10, \dots\)
- \(9, 5, 1, -3, -7, \dots\)
- Answer
-
- geometric \(a_{n}=54 \cdot\left(-\dfrac{1}{3}\right)^{n-1}\)
- neither
- arithmetic \(a_{n}=9-2 \cdot(n-1)\)
Find the sum of the first \(75\) terms of the arithmetic sequence: \[-30, -22, -14, -6, 2, \dots \nonumber \]
- Answer
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\(19950\)
Find the sum of the first \(8\) terms of the geometric series: \[-7, -14, -28, -56, -112, \dots \nonumber \]
- Answer
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\(-1785\)
Find the value of the infinite geometric series: \[80-20+5-1.25+\dots \nonumber \]
- Answer
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\(64\)
Expand the expression via the binomial theorem. \[(3x^2-2xy)^3 \nonumber \]
- Answer
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\(27 x^{6}-54 x^{5} y+36 x^{4} y^{2}-8 x^{3} y^{3}\)
Write the first \(3\) terms of the binomial expansion: \[\left(ab^2+\dfrac{10}{a}\right)^9 \nonumber \]
- Answer
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\(a^{9} b^{18}+90 a^{7} b^{16}+3600 a^{5} b^{14}\)
Find the \(6\)th term of the binomial expansion: \[(5p-q^2)^8 \nonumber\]
- Answer
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\(-7000 p^{3} q^{10}\)