27.5: Review of complex numbers, sequences, and the binomial theorem
- Page ID
- 68486
Multiply and write the answer in standard form: \[(-4)(\cos(207^\circ)+i\sin(207^\circ))\cdot 2(\cos(33^\circ)+i\sin(33^\circ)) \nonumber \]
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\(4+i 4 \sqrt{3}\)
Divide and write the answer in standard form: \[\dfrac{3(\cos(\frac{\pi}{3})+i\sin(\frac{\pi}{3}))}{15(\cos(\frac{\pi}{2})+i\sin(\frac{\pi}{2}))} \nonumber \]
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\(\dfrac{\sqrt{3}}{10}-i \dfrac{1}{10}\)
Find the magnitude and directional angle of the vector \[\vec{v}=\langle -7, -7\sqrt{3}\rangle \nonumber\]
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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.
- \(54, -18, 6, -2, \dfrac 2 3, \dots\)
- \(2, 4, 8, 10, \dots\)
- \(9, 5, 1, -3, -7, \dots\)
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- 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: \[-30, -22, -14, -6, 2, \dots \nonumber \]
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\(19950\)
Find the sum of the first \(8\) terms of the geometric series: \[-7, -14, -28, -56, -112, \dots \nonumber \]
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\(-1785\)
Find the value of the infinite geometric series: \[80-20+5-1.25+\dots \nonumber \]
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\(64\)
Expand the expression via the binomial theorem. \[(3x^2-2xy)^3 \nonumber \]
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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: \[\left(ab^2+\dfrac{10}{a}\right)^9 \nonumber \]
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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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\(-7000 p^{3} q^{10}\)