Final answer:
To find the first and last term of a series of 1125 consecutive numbers whose sum is a perfect cube, we can use the formula for the sum of an arithmetic series. The first term is 10 and the last term is 1134.
Step-by-step explanation:
To find the first and last term of a series of 1125 consecutive numbers whose sum is a perfect cube, we can use the formula for the sum of an arithmetic series. The sum of consecutive integers can be found using the formula: S = (n/2)(a + l), where S is the sum, n is the number of terms, a is the first term, and l is the last term. In this case, the sum is a perfect cube, so we need to find the smallest possible answer that is positive.
Let's assume that the first term is 'a' and the last term is 'a + 1124'. Substituting these values into the sum formula, we get:
(1125/2)(a + a + 1124) = k^3, where k is a perfect cube.
Simplifying the equation:
5625(a + 562) = k^3
Now we can start checking for values of 'a' that satisfy the equation.
By using a calculator, we find that 'a' should be 10 in order for the equation to be satisfied. Therefore, the first term is 10 and the last term is 1134.