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

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## Exercise $$\PageIndex{1}$$

Multiply and write the answer in standard form: $(-4)(\cos(207^\circ)+i\sin(207^\circ))\cdot 2(\cos(33^\circ)+i\sin(33^\circ)) \nonumber$

$$4+i 4 \sqrt{3}$$

## Exercise $$\PageIndex{2}$$

Divide and write the answer in standard form: $\dfrac{3(\cos(\frac{\pi}{3})+i\sin(\frac{\pi}{3}))}{15(\cos(\frac{\pi}{2})+i\sin(\frac{\pi}{2}))} \nonumber$

$$\dfrac{\sqrt{3}}{10}-i \dfrac{1}{10}$$

## Exercise $$\PageIndex{3}$$

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

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. $$9, 5, 1, -3, -7, \dots$$
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: $-30, -22, -14, -6, 2, \dots \nonumber$

$$19950$$

## Exercise $$\PageIndex{6}$$

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

$$-1785$$

## Exercise $$\PageIndex{7}$$

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

$$64$$

## Exercise $$\PageIndex{8}$$

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

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

## Exercise $$\PageIndex{9}$$

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

$$a^{9} b^{18}+90 a^{7} b^{16}+3600 a^{5} b^{14}$$

## Exercise $$\PageIndex{10}$$

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

$$-7000 p^{3} q^{10}$$