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27.5: Review of complex numbers, sequences, and the binomial theorem

  • Page ID
    68486
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    Exercise \(\PageIndex{1}\)

    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

    \(4+i 4 \sqrt{3}\)

    Exercise \(\PageIndex{2}\)

    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

    \(\dfrac{\sqrt{3}}{10}-i \dfrac{1}{10}\)

    Exercise \(\PageIndex{3}\)

    Find the magnitude and directional angle of the vector \[\vec{v}=\langle -7, -7\sqrt{3}\rangle \nonumber\]

    Answer

    magnitude \(\|\langle-7,-7 \sqrt{3}\rangle\|=14\), directional angle \(\theta=\frac{4 \pi}{3}=240^{\circ}\)

    Exercise \(\PageIndex{4}\)

    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.

    1. \(54, -18, 6, -2, \dfrac 2 3, \dots\)
    2. \(2, 4, 8, 10, \dots\)
    3. \(9, 5, 1, -3, -7, \dots\)
    Answer
    1. geometric \(a_{n}=54 \cdot\left(-\dfrac{1}{3}\right)^{n-1}\)
    2. neither
    3. arithmetic \(a_{n}=9-2 \cdot(n-1)\)

    Exercise \(\PageIndex{5}\)

    Find the sum of the first \(75\) terms of the arithmetic sequence: \[-30, -22, -14, -6, 2, \dots \nonumber \]

    Answer

    \(19950\)

    Exercise \(\PageIndex{6}\)

    Find the sum of the first \(8\) terms of the geometric series: \[-7, -14, -28, -56, -112, \dots \nonumber \]

    Answer

    \(-1785\)

    Exercise \(\PageIndex{7}\)

    Find the value of the infinite geometric series: \[80-20+5-1.25+\dots \nonumber \]

    Answer

    \(64\)

    Exercise \(\PageIndex{8}\)

    Expand the expression via the binomial theorem. \[(3x^2-2xy)^3 \nonumber \]

    Answer

    \(27 x^{6}-54 x^{5} y+36 x^{4} y^{2}-8 x^{3} y^{3}\)

    Exercise \(\PageIndex{9}\)

    Write the first \(3\) terms of the binomial expansion: \[\left(ab^2+\dfrac{10}{a}\right)^9 \nonumber \]

    Answer

    \(a^{9} b^{18}+90 a^{7} b^{16}+3600 a^{5} b^{14}\)

    Exercise \(\PageIndex{10}\)

    Find the \(6\)th term of the binomial expansion: \[(5p-q^2)^8 \nonumber\]

    Answer

    \(-7000 p^{3} q^{10}\)


    This page titled 27.5: Review of complex numbers, sequences, and the binomial theorem is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by Thomas Tradler and Holly Carley (New York City College of Technology at CUNY Academic Works) via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.