27.5: Revisión de números complejos, secuencias y teorema binomial
- Page ID
- 117668
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Multiplica y escribe la respuesta en forma estándar:\[(-4)(\cos(207^\circ)+i\sin(207^\circ))\cdot 2(\cos(33^\circ)+i\sin(33^\circ)) \nonumber \]
- Contestar
-
\(4+i 4 \sqrt{3}\)
Divide y escribe la respuesta en forma estándar:\[\dfrac{3(\cos(\frac{\pi}{3})+i\sin(\frac{\pi}{3}))}{15(\cos(\frac{\pi}{2})+i\sin(\frac{\pi}{2}))} \nonumber \]
- Contestar
-
\(\dfrac{\sqrt{3}}{10}-i \dfrac{1}{10}\)
Encuentra la magnitud y el ángulo direccional del vector\[\vec{v}=\langle -7, -7\sqrt{3}\rangle \nonumber\]
- Contestar
-
magnitud\(\|\langle-7,-7 \sqrt{3}\rangle\|=14\), ángulo direccional\(\theta=\frac{4 \pi}{3}=240^{\circ}\)
Determinar si la secuencia es una secuencia aritmética o geométrica o ninguna. Si es uno de estos, entonces encuentre la fórmula general para el término\(n\) th\(a_n\) de la secuencia.
- \(54, -18, 6, -2, \dfrac 2 3, \dots\)
- \(2, 4, 8, 10, \dots\)
- \(9, 5, 1, -3, -7, \dots\)
- Contestar
-
- geométrico\(a_{n}=54 \cdot\left(-\dfrac{1}{3}\right)^{n-1}\)
- ni
- aritmética\(a_{n}=9-2 \cdot(n-1)\)
Encuentra la suma de los primeros\(75\) términos de la secuencia aritmética:\[-30, -22, -14, -6, 2, \dots \nonumber \]
- Contestar
-
\(19950\)
Encuentra la suma de los primeros\(8\) términos de la serie geométrica:\[-7, -14, -28, -56, -112, \dots \nonumber \]
- Contestar
-
\(-1785\)
Encuentra el valor de la serie geométrica infinita:\[80-20+5-1.25+\dots \nonumber \]
- Contestar
-
\(64\)
Expandir la expresión a través del teorema binomial. \[(3x^2-2xy)^3 \nonumber \]
- Contestar
-
\(27 x^{6}-54 x^{5} y+36 x^{4} y^{2}-8 x^{3} y^{3}\)
Escribe los primeros\(3\) términos de la expansión binomial:\[\left(ab^2+\dfrac{10}{a}\right)^9 \nonumber \]
- Contestar
-
\(a^{9} b^{18}+90 a^{7} b^{16}+3600 a^{5} b^{14}\)
Encuentra el término\(6\) th de la expansión binomial:\[(5p-q^2)^8 \nonumber\]
- Contestar
-
\(-7000 p^{3} q^{10}\)