In an arithmetic progression, the sum of the first 11 terms is required, given that the first term is -31 and the last (11th) term is 29. What is the sum of these first 11 terms?

Introduction / Context:
This problem uses the sum formula for an arithmetic progression when the first and last terms are known. It is a very standard type of series question in aptitude tests and helps learners become comfortable with quickly computing sums without listing all individual terms.

Given Data / Assumptions:

• The sequence is an arithmetic progression.
• The first term, T1, is -31.
• The last term (which is T11) is 29.
• The number of terms n is 11.
• We need to compute the sum of the first 11 terms, S11.

Concept / Approach:
For an arithmetic progression, the sum of the first n terms is given by Sn = n / 2 * (first term + last term), when the first and nth terms are known. This formula comes from pairing terms from the beginning and end of the sequence, each pair having the same sum. Because both the first and the 11th term are given, we can apply the formula directly without first finding the common difference.

Step-by-Step Solution:
Step 1: Identify the given values: n = 11, first term a = -31, and last term l = 29.Step 2: Use the sum formula for an arithmetic progression: Sn = n / 2 * (a + l).Step 3: Substitute the values: S11 = 11 / 2 * (-31 + 29).Step 4: Compute the sum inside the parentheses: -31 + 29 = -2.Step 5: So S11 = 11 / 2 * (-2).Step 6: Simplify: 11 / 2 * (-2) = 11 * (-1) = -11.

Verification / Alternative check:
We can verify by noting that the average of the first and last term is (a + l) / 2 = (-31 + 29) / 2 = -2 / 2 = -1. In an arithmetic progression, the average of the first and last term equals the average of all terms when they are evenly spaced. Therefore, the average of all 11 terms is -1. Then the total sum is average * number of terms = -1 * 11 = -11. This matches the result from the formula, confirming that the answer is correct.

Why Other Options Are Wrong:
Option 42: This would correspond to a positive average value, which contradicts the fact that the first term is strongly negative and the last term is only slightly positive.
Option 28: Also suggests a positive average, inconsistent with the endpoints.
Option 12: This does not match the required average when consistent with the first and last terms.
Option 0: Would require the average of the sequence to be 0, which would need symmetric terms around 0, but -31 and 29 are not symmetric around 0.

Common Pitfalls:
Some learners forget that the sum formula uses (a + l) and mistakenly use only the first term or only the last term. Others try to compute all 11 terms explicitly, which is unnecessary and time-consuming. It is also easy to miscalculate -31 + 29 and accidentally write +2 instead of -2, which would completely change the answer. Careful handling of negative numbers is important.

Final Answer:
The sum of the first 11 terms of the arithmetic progression is -11.