Question

O valor da soma S: √4 + 1/√2 + 1 + 1/ √3 + √2 + 1/√4 + √3 +...+ 1/√196 + √195 é um número:
A) Natural menor que 10.
B) Natural maior que 10.
C) Racional não inteiro.
D) Irracional.​

Answer 1

Resposta:

S=√4 + 1/√2 + 1 + 1/ √3 + √2 + 1/√4 + √3 +...+ 1/√196 + √195

2

+

(√2 - 1)/(√2 + 1)(√2- 1) =(√2 - 1)/(2-1)=(√2 - 1)

+

(√3 -√2)/(√3 +√2)(√3 -√2) =(√3 -√2)/(3-2)=(√3 -√2)

+

(√4 -√3)/(√4 +√3)(√4 -√3) =(√4 -√3)/(3-2)=(√4 -√3)

+

.

.

+

(√194-√193)/(√194+√193)(√194 -√193) =(√194 -√193)/(194-193)=(√194 -√193)

+

(√195-√194)/(√195+√194)(√195 -√194) =(√195 -√194)/(195-194)=(√195 -√194)

+

(√196-√195)/(√196+√195)(√196 -√195) =(√196 -√195)/(196-195)=(√196 -√195)

S= 2 + √2 - 1 +√3 -√2 + √4 -√3 + .....+√194 -√193+√195 -√194+ √196 -√195

S=2 -1 + √196 = 1+14= 15

B) Natural maior que 10

Answer 2

Resposta:

$\boxed{\mathtt{B}}$

Explicação passo-a-passo:

$\\ \displaystyle \mathsf{S = \sqrt{4} + \frac{1}{\sqrt{2} + 1} + \frac{1}{\sqrt{3} + \sqrt{2}} + ... + \frac{1}{\sqrt{196} + \sqrt{195}}} \\\\\\ \mathsf{S = \sqrt{4} + \frac{1}{\sqrt{2} + \sqrt{1}} + \frac{1}{\sqrt{3} + \sqrt{2}} + ... + \frac{1}{\sqrt{196} + \sqrt{195}}} \\\\\\ \mathsf{S = \sqrt{4} + \sum_{n = 2}^{196} \frac{1}{\sqrt{n} + \sqrt{n - 1}}}$

Racionalizando,

$\displaystyle \mathtt{\frac{1}{\sqrt{n} + \sqrt{n - 1}} \cdot \frac{\sqrt{n} - \sqrt{n - 1}}{\sqrt{n} - \sqrt{n - 1}} = \frac{\sqrt{n} - \sqrt{n - 1}}{n - (n - 1)} = \boxed{\mathtt{\sqrt{n} - \sqrt{n - 1}}}}$

Com efeito,

$\\ \displaystyle \mathsf{S = \sqrt{4} + \sum_{n = 2}^{196} \frac{1}{\sqrt{n} + \sqrt{n - 1}}} \\\\\\ \mathsf{S = 2 + \sum_{n = 2}^{196} \sqrt{n} - \sqrt{n - 1}} \\\\\\ \mathsf{S = 2 + \sum_{n = 2}^{196} \sqrt{n} - \sum_{n = 2}^{196} \sqrt{n - 1}} \\\\\\ \mathsf{S = 2 + \left ( \sqrt{2} + \sqrt{3} + ... + \sqrt{195} + \sqrt{196} \right ) - \left ( \sqrt{1} + \sqrt{2} + ... + \sqrt{195} \right )}$

$\\ \displaystyle \mathsf{S = 2 + \left ( \sqrt{2} + \sqrt{3} + ... + \sqrt{195} + \sqrt{196} \right ) - \left ( \sqrt{1} + \sqrt{2} + ... + \sqrt{195} \right )} \\\\ \mathsf{S = 2 + \left ( \sqrt{196} - \sqrt{1} \right )} \\\\ \mathsf{S = 2 + (14 - 1)} \\\\ \boxed{\boxed{\mathsf{S = 15}}}$