Question

Разность кубов двух последовательных натуральных чисел равна 331. Чему равен куб суммы этих чисел?

Answer 1

Задание № 1:

Разность кубов двух последовательных натуральных чисел равна 331. Чему равен куб суммы этих чисел?

(x+1)^3-x^3=331

x^3+3x^2+3x+1-x^3=331

3x^2+3x-330=0

x^2+x-110=0

D=1+440=441

x=(-1-21)/2 - не натуральное

x=(-1+21)/2=10

х+1=11

(10+11)^3=9261

ответ: 9261

Answer 2

Problem Analysis

We are given that the difference between the cubes of two consecutive natural numbers is equal to 331. We need to find the cube of the sum of these two numbers.

Solution

Let's assume the two consecutive natural numbers are n and n+1.

According to the problem statement, the difference between the cubes of these two numbers is 331. Mathematically, we can represent this as:

(n+1)^3 - n^3 = 331

Expanding the cubes, we get:

n^3 + 3n^2 + 3n + 1 - n^3 = 331

Simplifying the equation, we have:

3n^2 + 3n + 1 = 331

Rearranging the terms, we get:

3n^2 + 3n - 330 = 0

Now, we have a quadratic equation in terms of n. We can solve this equation to find the value of n.

Using the quadratic formula, we have:

n = (-b ± √(b^2 - 4ac)) / (2a)

where a = 3, b = 3, and c = -330.

Solving this equation, we find two possible values for n.

Let's calculate the values of n:

n = (-3 ± √(3^2 - 4 * 3 * -330)) / (2 * 3)

n = (-3 ± √(9 + 3960)) / 6

n = (-3 ± √(3969)) / 6

n = (-3 ± 63) / 6

So, we have two possible values for n: n = 10 and n = -11.

Since we are dealing with natural numbers, we can discard the negative value of n.

Therefore, n = 10.

Now, we can find the value of n+1:

n+1 = 10 + 1 = 11

The two consecutive natural numbers are 10 and 11.

To find the cube of the sum of these numbers, we calculate:

(n + n+1) ^ 3 = (10 + 11) ^ 3 = 21 ^ 3

Calculating the cube, we get:

21 ^ 3 = 9261

Answer

The cube of the sum of the two consecutive natural numbers whose difference in cubes is equal to 331 is 9261.