Question

2*2019/((1+1/(1+2)+1/(1+2+3)+...+1/(1+2+3+...+2019)))

Answer 1

Ответ:

2020

Пошаговое объяснение:

$\dfrac{1}{1+2+...+k}=\dfrac{1}{\frac{1+k}{2}k}=\dfrac{2}{k(k+1)}$

Тогда:$\dfrac{2*2019}{1+\frac{1}{1+2}+...+\frac{1}{1+...+2019}}=\dfrac{2*2019}{\frac{2}{1*2}+\frac{2}{2*3}+...+\frac{2}{2019*2020}}=\dfrac{2019}{\frac{1}{1*2}+\frac{1}{2*3}+...+\frac{1}{2019*2020}}=\dfrac{2019}{\frac{1}{1}-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+...+\frac{1}{2019}-\frac{1}{2020}}=\dfrac{2019}{1-\frac{1}{2020}}=\dfrac{2019}{\frac{2019}{2020}}=2020$

Answer 2

To solve the given expression:

2 * 2019 / (1 + 1/(1 + 2) + 1/(1 + 2 + 3) + ... + 1/(1 + 2 + 3 + ... + 2019))

Let's break it down step by step:

Step 1: Simplify the denominator

The denominator is a sum of fractions with increasing denominators. We can simplify it by finding a pattern.

The first term is 1/(1 + 2), which simplifies to 1/3.

The second term is 1/(1 + 2 + 3), which simplifies to 1/6.

The third term is 1/(1 + 2 + 3 + 4), which simplifies to 1/10.

We can observe that the denominator follows the pattern of 2n + 1, where n is the number of terms.

So, the denominator can be written as:

1/3 + 1/6 + 1/10 + ... + 1/(2n + 1)

Step 2: Find the value of n

To find the value of n, we need to determine the number of terms in the denominator.

The last term in the denominator is 1/(1 + 2 + 3 + ... + 2019).

To find the sum of consecutive numbers from 1 to 2019, we can use the formula for the sum of an arithmetic series:

Sum = (n/2)(first term + last term)

In this case, the first term is 1 and the last term is 2019.

Sum = (2019/2)(1 + 2019) = 2039190

So, the number of terms, n, is 2019.

Step 3: Evaluate the expression

Now that we have the value of n, we can substitute it into the expression:

2 * 2019 / (1 + 1/3 + 1/6 + 1/10 + ... + 1/(2n + 1))

2 * 2019 / (1 + 1/3 + 1/6 + 1/10 + ... + 1/4039)

Step 4: Calculate the sum of the fractions

To calculate the sum of the fractions, we can use the formula for the sum of a harmonic series:

Sum = 1/1 + 1/2 + 1/3 + ... + 1/n

The sum of the fractions can be approximated using the natural logarithm:

Sum ≈ ln(n) + γ, where γ is the Euler-Mascheroni constant.

Using this approximation, we can calculate the sum of the fractions:

Sum ≈ ln(4039) + γ ≈ 8.303 + 0.577 ≈ 8.880

Step 5: Substitute the values and calculate the final result

Now, we can substitute the values into the expression:

2 * 2019 / (1 + 1/3 + 1/6 + 1/10 + ... + 1/4039)

= 2 * 2019 / (1 + 8.880)

= 2 * 2019 / 9.880

= 4078 / 9.880

≈ 412.95

Therefore, the value of the given expression is approximately 412.95.

Please note that the above calculation is an approximation, and the actual value may differ slightly due to the use of the natural logarithm approximation.

I hope this helps! Let me know if you have any further questions.