Final answer:
By taking an algebraic approach, it is shown that there is no positive integer n such that n³ + 6n equals 125, because substitution and logical reasoning with regards to the cubic values establish that no such positive integer can satisfy the given equation.
Step-by-step explanation:
To prove that there is no positive integer n such that n³ + 6n is equal to 125, we can approach the problem algebraically. Let's start by assuming that there exists a positive integer n that satisfies the equation n³ + 6n = 125. We can rewrite the given equation as n³ + 6n - 125 = 0.
To find a solution for n, we would typically try to factor the left side. However, since 125 is a cube of 5, we could check if n can be 5. Substituting 5 into the equation, we have (5)³ + 6(5) which equals 125 + 30, and this is clearly not equal to 125. This quick substitution shows that 5, the cube root of 125, is not a solution.
Moreover, any other positive integer value of n that is larger than 5 will result in a value for n³ that is larger than 125, and adding 6n to it will only increase that value. On the other hand, any positive integer value less than 5 will have an n³ less than 125. After adding 6n, the total would still be less than 125. Therefore, there is no positive integer n such that n³ + 6n equals 125.