Final answer:
To find the least integer x so that x, x^5, and 2x-15 form an arithmetic sequence, we set up an equation based on the constant difference property of arithmetic sequences and test integer factors of 15 to find the solution x = 1.
Step-by-step explanation:
To find the least integer x for which x, x^5, and 2x-15 may represent three consecutive terms of an arithmetic sequence, we need to use the property that the difference between consecutive terms in an arithmetic sequence is constant. Therefore, the difference between the second and the first term should be the same as the difference between the third and the second term. We can express this as:
x^5 - x = (2x - 15) - x^5
By simplifying the equation, we get:
2x^5 - 2x + 15 = 0
This is a polynomial equation in x which is not easy to solve algebraically. However, we can search for integer solutions by checking the possible integer values of x that make the polynomial equal to zero, considering that an integer solution has to be a factor of the constant term, which in this case is 15.
By testing potential integer factors of 15 (±1, ±3, ±5, ±15), we find that the smallest integer value that satisfies the equation is x = 1.