Final answer:
The sum of the series 17+27+37+...+417 can be calculated using the formula for the sum of an arithmetic series after finding the number of terms through the formula for the nth term of an arithmetic sequence.
Step-by-step explanation:
The sum of the series 17+27+37+...+417 is found by identifying the series as an arithmetic sequence, where each term increases by a constant difference. In this case, the common difference is 10. We use the formula for the sum of an arithmetic series, which is S = n/2(a1 + an), where S is the sum of the series, n is the number of terms, a1 is the first term, and an is the last term.
To find n, we can use the formula for the nth term of an arithmetic sequence: an = a1 + (n - 1)d, where d is the common difference. Re-arranging this formula to solve for n, we get n = ((an - a1) / d) + 1.
Applying these to our series: the first term a1 is 17, the last term an is 417, and the common difference d is 10. This gives us n = ((417 - 17) / 10) + 1 = 41. Substituting the values into the sum formula, we have S = 41/2(17 + 417), and evaluating this gives us the sum of the arithmetic series.