The sum of the first 17 terms of an arithmetic progression is required. If the first term is -20 and the 17th (last) term is 28, what is this sum?

Introduction / Context:
This question is about summing the terms of an arithmetic progression (A.P.). You are told the first term and the 17th term of an A.P. and asked to find the sum of the first 17 terms. This is a standard application of the formula for the sum of the first n terms of an A.P. and tests your understanding of series formulas and basic arithmetic.

Given Data / Assumptions:

- The number of terms n is 17. - The first term a is -20. - The 17th term, which is also the last term in the sum, is 28. - We need the sum S_17 of the first 17 terms.

Concept / Approach:
For an arithmetic progression, the sum S_n of the first n terms can be computed using the formula S_n = (n / 2) * (first term + last term). This works because pairing the first and last terms, second and second last terms and so on produces n / 2 pairs, each with the same sum. In this question, the first and 17th terms are given directly, so we can apply the formula without needing to find the common difference explicitly.

Step-by-Step Solution:
Step 1: Note the values: n = 17, a_1 = -20 and a_17 = 28. Step 2: Use the sum formula for an A.P.: S_n = (n / 2) * (first term + last term). Step 3: Substitute n = 17, first term = -20 and last term = 28 into the formula: S_17 = (17 / 2) * (-20 + 28). Step 4: Compute the sum inside the brackets: -20 + 28 = 8. Step 5: So S_17 = (17 / 2) * 8. Step 6: Simplify (17 / 2) * 8. First compute 8 / 2 = 4, then 17 * 4 = 68. Step 7: Therefore the sum of the first 17 terms is 68.

Verification / Alternative check:
We can also verify by finding the common difference and checking a few terms. The common difference d can be found from the formula for the n-th term: a_17 = a_1 + (17 - 1)d. Substitute 28 = -20 + 16d, which gives 16d = 48 and hence d = 3. The sequence starts as -20, -17, -14, ..., and increases by 3 each time. Using the sum formula with n = 17, a_1 = -20 and a_17 = 28 still gives S_17 = 17 * ( -20 + 28 ) / 2 = 68, confirming the earlier result.

Why Other Options Are Wrong:
Values 156, 142 and 242 result from miscalculations such as adding the first and last terms incorrectly, using the wrong number of terms or forgetting to divide by 2 in the sum formula. None of these values match the correctly computed sum of 68. Only 68 is consistent with both the formula and the specific terms of the progression.

Common Pitfalls:
A frequent mistake is to forget the division by 2 in the formula S_n = n * (first + last) / 2 or to use the wrong last term when n is given. Some learners also incorrectly assume the last term is n multiplied by the common difference without checking the given value. Always confirm the correct first and last terms and carefully follow the formula steps to avoid such errors.

Final Answer:
The sum of the first 17 terms of the progression is 68, which corresponds to option A.