In an arithmetic progression, the fourth term is 11 and the seventh term is -4. What is the sum of the first 11 terms of this arithmetic progression?

Introduction / Context:
This question tests your ability to use information about two nonconsecutive terms of an arithmetic progression to find both the first term and the common difference, and then compute the sum of a given number of terms. It is slightly more involved than a direct sum question, so it is a good test of conceptual understanding of arithmetic sequences.

Given Data / Assumptions:

• Let the first term of the arithmetic progression be a.
• Let the common difference be d.
• The fourth term a4 is 11.
• The seventh term a7 is -4.
• We must find the sum of the first 11 terms, S_11.

Concept / Approach:
We use the nth term formula of an arithmetic progression: a_n = a + (n - 1) * d. From the given fourth term and seventh term, we build two equations in a and d. Solving these equations gives us a and d. Once we know a and d, we can either apply the sum formula S_n = n * (2a + (n - 1) * d) / 2 or S_n = n * (a1 + a_n) / 2 to find S_11. This approach combines equation solving and use of the sum formula.

Step-by-Step Solution:
Write the fourth term: a4 = a + 3d = 11. Write the seventh term: a7 = a + 6d = -4. Subtract the first equation from the second: (a + 6d) - (a + 3d) = -4 - 11. This simplifies to 3d = -15, so d = -15 / 3 = -5. Substitute d = -5 into a + 3d = 11. Then a + 3 * (-5) = 11, so a - 15 = 11. Therefore a = 11 + 15 = 26. Now use the sum formula S_n = n * (2a + (n - 1) * d) / 2 for n = 11. Compute S_11 = 11 * (2 * 26 + 10 * (-5)) / 2. Inside the brackets: 2 * 26 = 52 and 10 * (-5) = -50, so 52 - 50 = 2. Thus S_11 = 11 * 2 / 2 = 11. So the sum of the first 11 terms is 11.

Verification / Alternative Check:
To verify, let us list several terms. With a = 26 and d = -5, the sequence begins 26, 21, 16, 11, 6, 1, -4, -9, -14, -19, -24. We can check the given data: the fourth term is 11 and the seventh term is -4, which matches. Now add the terms in pairs from both ends: (26 + -24) = 2, (21 + -19) = 2, (16 + -14) = 2, (11 + -9) = 2, (6 + -4) = 2 and the middle term is 1. So we have five pairs each summing to 2 plus the middle 1. The total sum is 5 * 2 + 1 = 10 + 1 = 11, confirming our earlier calculation.

Why Other Options Are Wrong:

• Option -75: This would correspond to an incorrect calculation of the sum or a wrong common difference, and it does not match the actual series sum.
• Option 55: This would arise if someone mistakenly treated the first term as 11 and assumed a simple pattern, but it does not satisfy the correct equations.
• Option 100: This is significantly larger than the correct value and is inconsistent with the decreasing nature of the sequence.
• Option -11: This would have the right magnitude but wrong sign and usually arises from carelessly pairing terms or misplacing negative signs.

Common Pitfalls:
Students often forget that the nth term formula uses (n - 1) * d, not n * d, which leads to incorrect equations for a4 and a7. Another common mistake is to mishandle negative differences, especially when substituting back to find the first term. When using the sum formula, it is easy to forget to multiply by the correct number of terms or to apply the negative sign incorrectly inside the brackets. Writing each step clearly and checking signs at each stage helps avoid these problems.

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