(ITA - 2022) $z = 5 - 5i \in \mathbb{C}$, $f(n) = \left| z^{(2n+1)} + \overline{z}^{\,(2n+1)} \right|$ para cada $n \in \mathbb{N}$. A soma de $f(n)$ para $n$ de 1 até 20 é:

$$z = |z|\operatorname{cis}\theta$$ $$z^{(2n+1)} = |z|^{(2n+1)}(\cos{(2n+1)\theta} + i\operatorname{sen}{(2n+1)\theta})$$ $$\overline{z}^{(2n+1)} = |z|^{(2n+1)}(\cos{(2n+1)\theta} - i\operatorname{sen}{(2n+1)\theta})$$

$$= \left| 2|z|^{(2n+1)}\cos{(2n+1)\theta} \right|$$

$$2 \sum_{n=1}^{20} {|z|^{(2n+1)} \left| \cos{(2n+1)\theta} \right|}$$

$$|z| = \sqrt{50} = 5\sqrt{2}$$ $$\cos\theta = \frac{5}{5\sqrt{2}} = \frac{1}{\sqrt{2}} = \frac{\sqrt{2}}{2}$$ $$\theta = 2\pi - \frac{\pi}{4} = \frac{7\pi}{4}$$

$$2 \sum_{n=1}^{20} \left| (5\sqrt{2})^{(2n+1)} \cos\left((2n+1)\frac{7\pi}{4}\right) \right|$$

$$(2n + 1)\frac{7\pi}{4} = \frac{7n\pi}{2} + \frac{7\pi}{4}$$ $$\cos\left(\frac{21\pi}{4}\right) = \cos\left(5\pi + \frac{\pi}{4}\right) = -\frac{\sqrt{2}}{2}$$ $$\cos\left(\frac{35\pi}{4}\right) = \cos\left(8\pi + \frac{3\pi}{4}\right) = -\frac{\sqrt{2}}{2}$$ $$\dots$$

$$\left| \cos\left((2n+1)\frac{7\pi}{4}\right) \right| = \frac{\sqrt{2}}{2}$$

$$2 \cdot \frac{\sqrt{2}}{2} \sum_{n=1}^{20} \left| (5\sqrt{2})^{(2n+1)} \right|$$

$$f(n) = \sqrt{2} (5\sqrt{2})^{2n+1} = \sqrt{2} \cdot 5^{2n+1} \cdot 2^{\frac{2n+1}{2}} = 5^{2n+1} \cdot 2^{\frac{2n+2}{2}} = 5^{2n+1} \cdot 2^{n+1} = 10 \cdot 5^{2n} \cdot 2^n = 10 \cdot 50^n$$

$$10 \sum_{n=1}^{20} 50^n = 10(50 + 50^2 + \dots + 50^{20}) = 10\left(50 \frac{50^{20} - 1}{50 - 1}\right) = \frac{500(50^{20} - 1)}{49}$$

Antes:

A soma de f(n) para n de 1 ate 20 e:

z = 5 - 5i em C, f(n) = |z^(2n+1) + conjugado(z)^(2n+1)| para cada n em N.

z = |z|cisθ
z^(2n+1) = |z|^(2n+1)(cos((2n+1)θ) + isen((2n+1)θ))
conjugado(z)^(2n+1) = |z|^(2n+1)(cos((2n+1)θ) - isen((2n+1)θ))

= |2|z|^(2n+1)cos((2n+1)θ)|

2 sum n=1..20 |z|^(2n+1) |cos((2n+1)θ)|

|z| = sqrt(50) = 5sqrt(2)
cos θ = 5 / (5sqrt(2)) = sqrt(2)/2
θ = 2pi - pi/4 = 7pi/4

2 sum n=1..20 |(5sqrt(2))^(2n+1) cos((2n+1)7pi/4)|

(2n + 1)7pi/4 = 7npi/2 + 7pi/4
cos(21pi/4) = cos(5pi + pi/4) = -sqrt(2)/2
cos(35pi/4) = cos(8pi + 3pi/4) = -sqrt(2)/2
...

|cos((2n+1)7pi/4)| = sqrt(2)/2

2 * sqrt(2)/2 sum n=1..20 |(5sqrt(2))^(2n+1)|

f(n) = sqrt(2) (5sqrt(2))^(2n+1)
     = sqrt(2) * 5^(2n+1) * 2^((2n+1)/2)
     = 5^(2n+1) * 2^((2n+2)/2)
     = 5^(2n+1) * 2^(n+1)
     = 10 * 5^(2n) * 2^n
     = 10 * 50^n

10 sum n=1..20 50^n
= 10(50 + 50^2 + ... + 50^20)
= 10(50(50^20 - 1)/(50 - 1))
= 500(50^20 - 1)/49

Antes:

| z^{(2n+1)}

z = |z|cis\theta

z^{(2n+1)} = |z|^{(2n+1)}(cos{(2n+1)theta} + isen{(2n+1)theta})
\overline{z}^{(2n+1)} = |z|^{(2n+1)}(cos{(2n-1)theta} - isen{(2n+1)theta})

| 2|z|^{(2n+1)}cos{(2n+1)theta}
2 \sum_k=1 ^20 {|z|^{(2n+1)} cos{(2n+1)theta}}
|z| = sqrt(50) = 5sqrt(2)
costheta = 5/5sqrt(2) = 1/sqrt(2) . sqrt(2)/sqrt(2) = sqrt(2)/2
theta = 2pi - pi/4 = 7pi/4
2 \sumk=1 ^20 |{5sqrt(2)^{(2n+1)} cos{(2n+1) 7pi/4}}|

(2n + 1) 7pi/4 = 7npi/2 + 7pi/4
cos(21pi/4) = cos(5pi + pi/4) = -sqrt(2)/2
cos(35pi/4) = cos(8pi + 3pi/4) = -sqrt(2)/2
...

| cos((2n+1)7pi/4) | = sqrt(2)/2
2.sqrt(2)/2 \sum k=1 ^20 |(5sqrt(2))^{(2n+1)}|

f(n) = sqrt(2) (5sqrt(2))^{2n+1} = sqrt(2) 5^(2n+1) 2^(2n+1)/2} = 5^(2n+1)
2^((2n+2)/2) = 5^(2n+1) + 2^(n+1) = 10 . 5^2n + 2^n = 50^n . 10

10 \sum k=1 ^20 50^n = 10(50 + 50^2 + ... + 50^20) = 10(50 (50^20 - 1)/(50 - 1))
= 500(50^20 - 1)/49