Question

por favor me ajudem plss

19da8cccb9202773ab03654c78454bbc.pdf

Answer 1

Primeiro vou lembrar alguns conceitos:

$i^1=\sqrt{-1}\\\\i^2 = -1\\\\i^3= -i\\\\i^4=1\\\\i^5 = i^1$

Forma algébrica:

$z = a + bi$

Agora o dever:

1)

$2i^2 +i^7 = -2 -i\\\\i^{16} - i^{43}= 1 - (-i) = 1+i\\\\5i^{29} + 2i^{31} = 5i -2i = 3i\\\\4i^{26} + 3i^{84}= -4 +3 =-1\\\\i^ {62} - 6i ^{9}= -1 -6i \\\\ i^3 + i^5 + i^7 = -i +i -i = -i\\\\i^{13} = i$

2)

$a) \: z = -3+5i \\\\b) \: z = 2x+4i\\\\c) \: z = 0+2i\\\\d) \: z = -5a-i$

3)

$z_1 = (-3,5) = -3 +5i\\\\z_2 = (-5,2)= -5+2i\\\\z_3 = (-1,-1) = -1 -i\\\\z_4= (2,-6) = 2-6i\\\\a) \\\\ z_1+z_2 = -3-5+5i+2i=-8+7i\\\\ z_2 +z_3 =-5+2i-1-i = -6+i\\\\ z_3+z_4=-1-i+2-6i = 1-7i\\\\z_4+z_5 = \text{nem tem } z_5\\\\z_2 +z_4=-5+2i+2-6i= -3 -4i$

4)

$S= i+i^2+i^3+ i^4 +i^5+\dots+ i^{306}$

Observe que o ciclo repete varias vezes:

$S = (i+i^2+i^3+i^4)+(i+i^2+i^3+i^4)+....+i^{306}\\\\S = (i+i^2+i^3+i^4)+(i+i^2+i^3+i^4)+....+i^{2}$

O ciclo repete até o i¹+i² no final.

$S=(0)+(0)+\dots + (0)+i^1+i^2\\\\S = i-1$

5)

$y= i+i^2+i^3+ i^4 +i^5+\dots+ i^{1001}\\\\y= (i+i^2+i^3+i^4)+(i+i^2+i^3+i^4)+\dots +(i+i^2+i^3+i^4)+ i\\\\y = (0)+(0)+\dots +(0)+i\\\\y=i$

6)

$\dfrac{2+3i}{1+2i} = \dfrac{(2+3i)(1-2i)}{1^2 -4i^2} = \dfrac{2-4i+3i-6i^2}{1+4}=\dfrac{8-i}{5}\\\\~\\ \dfrac{1}{3+2i} = \dfrac{3-2i}{9-4i^2}=\dfrac{3-2i}{9+4} = \dfrac{3-2i}{13}\\\\~\\ \dfrac{1+3i}{1-i} = \dfrac{(1+3i)(1+i)}{1^2-i^2} = \dfrac{1+i +3i+3i^2}{1+1} = \dfrac{4i-2}{2} = 2i-1$