Question

(1÷(√5+√2))+(1÷(√8+√5))+(1÷(√11+√8))+...+(1÷(√38+√35)) Найти значение выражения.

Answer 1

$\sf \dfrac{1}{\sqrt{5}+\sqrt{2}}+\dfrac{1}{\sqrt{8}+\sqrt{5}}+\dfrac{1}{\sqrt{11}+\sqrt{8}}+\ \cdot \cdot \cdot \ +\dfrac{1}{\sqrt{38}+\sqrt{35}}=$

Каждую дробь домножаем на сопряженное.

$\sf =\dfrac{\sqrt{5}-\sqrt{2}}{5-2}+\dfrac{\sqrt{8}-\sqrt{5}}{8-5}+\dfrac{\sqrt{11}-\sqrt{8}}{11-8}+ \ \cdot \cdot \cdot \ +\dfrac{\sqrt{38}-\sqrt{35}}{38-35}=$

Заметим, что в знаменателе каждой из дробей получается число 3, а корни в числителях взаимоуничтожаются, оставляя лишь крайние -√2 и √38.

$\sf =\dfrac{-\sqrt{2}+\sqrt{38}}{3}=\dfrac{\sqrt{38}-\sqrt{2}}{3}$

Это окончательный ответ.

Answer 2

To find the value of the given expression:

(1/(√5+√2)) + (1/(√8+√5)) + (1/(√11+√8)) + ... + (1/(√38+√35))

We can start by simplifying each individual fraction. The trick here is to multiply each fraction by the conjugate of the denominator to eliminate the square root in the denominator. The conjugate of √a + √b is √a - √b.

Let's do it step by step:

1. For the first fraction, (1/(√5+√2)), multiply the numerator and denominator by the conjugate, (√5 - √2):

(1/(√5+√2)) * ((√5 - √2)/(√5 - √2)) = (√5 - √2) / ((√5)^2 - (√2)^2) = (√5 - √2) / (5 - 2) = (√5 - √2) / 3

2. For the second fraction, (1/(√8+√5)), multiply the numerator and denominator by the conjugate, (√8 - √5):

(1/(√8+√5)) * ((√8 - √5)/(√8 - √5)) = (√8 - √5) / ((√8)^2 - (√5)^2) = (√8 - √5) / (8 - 5) = (√8 - √5) / 3

3. For the third fraction, (1/(√11+√8)), multiply the numerator and denominator by the conjugate, (√11 - √8):

(1/(√11+√8)) * ((√11 - √8)/(√11 - √8)) = (√11 - √8) / ((√11)^2 - (√8)^2) = (√11 - √8) / (11 - 8) = (√11 - √8) / 3

We can see that each fraction has the same value, which is (√n+1 - √n) / 3, where n is the number inside the square root (√n).

Now, we need to find the value of this expression for n = 2, 5, 8, ..., 35.

Let's calculate:

n = 2: (√2+1 - √2) / 3 = (1/3) n = 5: (√5+1 - √5) / 3 = (1/3) n = 8: (√8+1 - √8) / 3 = (1/3) ... n = 35: (√35+1 - √35) / 3 = (1/3)

Now, we can see that the value of each term is (1/3), and there are a total of 12 terms (from n = 2 to n = 35).

So, the final value of the expression is:

(1/3) + (1/3) + (1/3) + ... + (1/3) (12 times)

= 12 * (1/3)

= 4

Therefore, the value of the expression is 4.