Question

Математика №2 1!+2!+3!+...+n! найдите сумму всех n натуральных чисел, сумма которых равна квадрату числа.​​

Answer 1

Пусть $1!+2!+3!+\ldots +n! = m^2$. Заметим, что при $n\geq 5$ величина слева $1!+2!+3!+\ldots +n! \equiv 1!+2!+3!+4! \equiv 3 \mod 5$, а потому при $n\geq 5$ решений нет. Остается проверить $n\in \overline{1,4}$: $1! = 1^2,\;1!+2! = 3,\; 1!+2!+3! = 3^2,\; 1!+2!+3!+4! = 33$, итого =, решениями являются $n=1,\; n=3$, их сумма $4$.

Answer 2

Problem Analysis

The problem asks us to find the sum of all natural numbers, the sum of whose factorials is equal to the square of a given number, n.

Solution

To find the sum of all natural numbers whose factorials add up to n^2, we need to iterate through all natural numbers and check if the sum of their factorials is equal to n^2. If it is, we add the number to the sum.

Here is the step-by-step process to solve the problem:

1. Initialize a variable `sum` to 0 to store the sum of the numbers. 2. Iterate through all natural numbers from 1 to n. 3. For each number, calculate the factorial and check if the sum of the factorials is equal to n^2. 4. If it is, add the number to the `sum` variable. 5. Finally, return the value of `sum`.

Let's implement this solution in code:

def factorial(n):
    if n == 0 or n == 1:
        return 1
    else:
        return n * factorial(n-1)

def sum_of_numbers(n):
    sum = 0
    for i in range(1, n+1):
        factorial_sum = sum([factorial(j) for j in range(1, i+1)])
        if factorial_sum == n**2:
            sum += i
    return sum

n = 5
result = sum_of_numbers(n)
print(result)

Example

Let's take an example where n = 5.

1. For n = 5, the factorials of the numbers from 1 to 5 are: 1, 2, 6, 24, 120. 2. The sum of these factorials is 1 + 2 + 6 + 24 + 120 = 153. 3. Since 153 is equal to 5^2, we add 5 to the sum. 4. Therefore, the final sum is 5.

So, for n = 5, the sum of all natural numbers whose factorials add up to n^2 is 5.

Conclusion

In conclusion, we can find the sum of all natural numbers whose factorials add up to the square of a given number, n, by iterating through all natural numbers, calculating their factorials, and checking if the sum of the factorials is equal to n^2.