In an arithmetic progression, the 3rd term is -14 and the 8th term is 1. What is the 11th term of this progression?

Introduction / Context:
This arithmetic progression (A.P.) problem gives you the 3rd and 8th terms and asks for the 11th term. Since every term in an A.P. differs from the previous one by a fixed common difference, two known terms allow you to determine both the first term and the common difference. Once these are known, you can compute any term, including the 11th. This question tests the systematic use of the general term formula for an A.P.

Given Data / Assumptions:

- Let the first term be a. - Let the common difference be d. - The 3rd term T_3 is -14. - The 8th term T_8 is 1. - We need to find the 11th term T_11.

Concept / Approach:
The general formula for the n-th term of an A.P. is T_n = a + (n - 1)d. Using this, we can write expressions for T_3 and T_8 in terms of a and d. These produce two linear equations that we solve to find a and d. Then we substitute these values into T_11 = a + 10d to obtain the 11th term. This method is consistent and avoids guesswork, even when terms are negative.

Step-by-Step Solution:
Step 1: Express T_3 using the general formula: T_3 = a + (3 - 1)d = a + 2d. Step 2: Given T_3 = -14, we have a + 2d = -14. Call this Equation (1). Step 3: Express T_8 using the general formula: T_8 = a + (8 - 1)d = a + 7d. Step 4: Given T_8 = 1, we have a + 7d = 1. Call this Equation (2). Step 5: Subtract Equation (1) from Equation (2) to eliminate a: (a + 7d) - (a + 2d) = 1 - (-14). Step 6: This leads to 5d = 15. Step 7: Solve for d: d = 15 / 5 = 3. Step 8: Substitute d = 3 back into Equation (1): a + 2 * 3 = -14, so a + 6 = -14. Step 9: Thus a = -14 - 6 = -20. Step 10: Now compute T_11 using T_11 = a + (11 - 1)d = a + 10d. Step 11: Substitute a = -20 and d = 3: T_11 = -20 + 10 * 3. Step 12: Compute 10 * 3 = 30, giving T_11 = -20 + 30 = 10.

Verification / Alternative check:
List some terms of the progression using a = -20 and d = 3. The sequence begins -20, -17, -14, -11, -8, -5, -2, 1, 4, 7, 10, ... The 3rd term is -14 and the 8th term is 1, matching the problem statement. Continuing, the 11th term in this list is 10, which agrees with our formula calculation. This confirms the correctness of the computed 11th term.

Why Other Options Are Wrong:
Values 14, 16 and 20 arise from incorrect computations of the common difference or miscounting the number of steps from a known term. Only 10 is consistent with the progression defined by T_3 = -14 and T_8 = 1 and the constant increase of 3 between consecutive terms.

Common Pitfalls:
Learners sometimes subtract equations in the wrong order, introducing sign errors in the value of d. Others may use the wrong n when computing T_11, using 11d instead of 10d. Remember that T_n = a + (n - 1)d, and always recheck the arithmetic when handling negative numbers to avoid accidental sign mistakes.

Final Answer:
The 11th term of the arithmetic progression is 10, which corresponds to option D.