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What is the largest of three consecutive natural numbers if the cube of the first number is smaller than their product by 5, the cube of the second number is greater than their product by 2 and the square of the third number exceeds their product by 3.
- a)3
- b)2
- c)1
- d)4
- e)None of the above
Correct answer is option 'A'. Can you explain this answer?
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Verified Answer
What is the largest of three consecutive natural numbers if the cube o...
The largest number cannot be 1 or 2 as is the largest of three consecutive natural numbers.
Now, consider the options.
If the largest number is 3, the numbers are 1,2 and 3. Their product is 6. It exceeds the cube of 1 by 5, is less than the cube of 2 by 2 and is less than the square of 3 by 3.
Thus 3 satisfies all the conditions.
Hence, option 1.
Now, consider the options.
If the largest number is 3, the numbers are 1,2 and 3. Their product is 6. It exceeds the cube of 1 by 5, is less than the cube of 2 by 2 and is less than the square of 3 by 3.
Thus 3 satisfies all the conditions.
Hence, option 1.
Most Upvoted Answer
What is the largest of three consecutive natural numbers if the cube o...
To solve this problem, let's assume the three consecutive natural numbers as x, x+1, and x+2.
Given conditions:
1. Cube of the first number is smaller than their product by 5:
x^3 = (x)(x+1)(x+2) - 5
2. Cube of the second number is greater than their product by 2:
(x+1)^3 = (x)(x+1)(x+2) + 2
3. Square of the third number exceeds their product by 3:
(x+2)^2 = (x)(x+1)(x+2) + 3
Now, let's solve these equations step by step to find the value of x.
Solving equation 1:
x^3 = (x)(x+1)(x+2) - 5
x^3 = (x^2 + 2x)(x+2) - 5
x^3 = x^3 + 2x^2 + 4x^2 + 4x - 5
0 = 6x^2 + 4x - 5
Solving equation 2:
(x+1)^3 = (x)(x+1)(x+2) + 2
(x+1)^3 = (x^2 + x)(x+2) + 2
(x+1)^3 = x^3 + 2x^2 + 2x^2 + 2x + 2
0 = -x^3 + 5x + 2
Solving equation 3:
(x+2)^2 = (x)(x+1)(x+2) + 3
(x+2)^2 = (x^2 + x)(x+2) + 3
(x+2)^2 = x^3 + 2x^2 + x^2 + 2x + 3
0 = -x^3 + 4x + 1
Now, we can solve these equations using any numerical method or by substituting the answer choices to find the value of x.
If we substitute x = 3, we get:
Equation 1: 3^3 = (3)(4)(5) - 5 => 27 = 60 - 5 => 27 = 55 (Not true)
Equation 2: (3+1)^3 = (3)(4)(5) + 2 => 64 = 60 + 2 => 64 = 62 (Not true)
Equation 3: (3+2)^2 = (3)(4)(5) + 3 => 25 = 60 + 3 => 25 = 63 (Not true)
If we substitute x = 2, we get:
Equation 1: 2^3 = (2)(3)(4) - 5 => 8 = 24 - 5 => 8 = 19 (Not true)
Equation 2: (2+1)^3 = (2)(3)(4) + 2 => 27 = 24 + 2 => 27 = 26 (Not true)
Equation 3: (2+2)^2 = (2)(3)(4) + 3 => 16 = 24 + 3
Given conditions:
1. Cube of the first number is smaller than their product by 5:
x^3 = (x)(x+1)(x+2) - 5
2. Cube of the second number is greater than their product by 2:
(x+1)^3 = (x)(x+1)(x+2) + 2
3. Square of the third number exceeds their product by 3:
(x+2)^2 = (x)(x+1)(x+2) + 3
Now, let's solve these equations step by step to find the value of x.
Solving equation 1:
x^3 = (x)(x+1)(x+2) - 5
x^3 = (x^2 + 2x)(x+2) - 5
x^3 = x^3 + 2x^2 + 4x^2 + 4x - 5
0 = 6x^2 + 4x - 5
Solving equation 2:
(x+1)^3 = (x)(x+1)(x+2) + 2
(x+1)^3 = (x^2 + x)(x+2) + 2
(x+1)^3 = x^3 + 2x^2 + 2x^2 + 2x + 2
0 = -x^3 + 5x + 2
Solving equation 3:
(x+2)^2 = (x)(x+1)(x+2) + 3
(x+2)^2 = (x^2 + x)(x+2) + 3
(x+2)^2 = x^3 + 2x^2 + x^2 + 2x + 3
0 = -x^3 + 4x + 1
Now, we can solve these equations using any numerical method or by substituting the answer choices to find the value of x.
If we substitute x = 3, we get:
Equation 1: 3^3 = (3)(4)(5) - 5 => 27 = 60 - 5 => 27 = 55 (Not true)
Equation 2: (3+1)^3 = (3)(4)(5) + 2 => 64 = 60 + 2 => 64 = 62 (Not true)
Equation 3: (3+2)^2 = (3)(4)(5) + 3 => 25 = 60 + 3 => 25 = 63 (Not true)
If we substitute x = 2, we get:
Equation 1: 2^3 = (2)(3)(4) - 5 => 8 = 24 - 5 => 8 = 19 (Not true)
Equation 2: (2+1)^3 = (2)(3)(4) + 2 => 27 = 24 + 2 => 27 = 26 (Not true)
Equation 3: (2+2)^2 = (2)(3)(4) + 3 => 16 = 24 + 3
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