In an arithmetic progression, the 2nd term is 17 and the 8th term is -1. What is the 14th term of this progression?

Introduction / Context:
This problem is about arithmetic progressions (A.P.), where each term after the first is obtained by adding a constant common difference. You are given the 2nd and 8th terms of an arithmetic progression and asked to find the 14th term. The question checks your ability to use the general formula for the n-th term of an A.P. and to solve for the first term and common difference using the given terms.

Given Data / Assumptions:

- Let the first term of the A.P. be a. - Let the common difference be d. - The 2nd term T_2 is 17. - The 8th term T_8 is -1. - We must find the 14th term T_14.

Concept / Approach:
The n-th term of an arithmetic progression is given by T_n = a + (n - 1)d. Using this formula, we can write equations for T_2 and T_8 in terms of a and d. Solving these two linear equations gives us the values of a and d. Once we know a and d, we substitute them into the formula for T_14 to obtain the required term. This method is standard for A.P. problems where two terms are given and another term is required.

Step-by-Step Solution:
Step 1: Write T_2 in terms of a and d: T_2 = a + (2 - 1)d = a + d. Step 2: We are told T_2 = 17, so a + d = 17. This is Equation (1). Step 3: Write T_8 in terms of a and d: T_8 = a + (8 - 1)d = a + 7d. Step 4: We are told T_8 = -1, so a + 7d = -1. This is Equation (2). Step 5: Subtract Equation (1) from Equation (2) to eliminate a: (a + 7d) - (a + d) = -1 - 17. Step 6: This gives 6d = -18, so d = -3. Step 7: Substitute d = -3 back into Equation (1): a + (-3) = 17, so a = 20. span style="display:block;">Step 8: Now find T_14 using T_14 = a + (14 - 1)d = a + 13d. Step 9: Substitute a = 20 and d = -3: T_14 = 20 + 13 * (-3) = 20 - 39. Step 10: Compute 20 - 39 = -19, so T_14 = -19.

Verification / Alternative check:
We can generate a few terms to check consistency. With a = 20 and d = -3, the sequence begins 20, 17, 14, 11, 8, 5, 2, -1, ... The 2nd term is indeed 17 and the 8th term is -1. Continuing this pattern, each term decreases by 3. From the 8th term -1 to the 14th term, there are 6 steps, each subtracting 3, giving -1 - 18 = -19. This matches our formula based calculation.

Why Other Options Are Wrong:
Values such as -22, -25 and -28 correspond to using incorrect values for the common difference or miscounting the steps when moving from the known term to the 14th term. Only -19 is consistent with both T_2 = 17 and T_8 = -1 and follows the correct arithmetic progression pattern with d = -3.

Common Pitfalls:
Some learners mistakenly use the formula T_n = a + nd instead of a + (n - 1)d, which shifts all terms by one position. Others may incorrectly subtract equations or mis-handle signs, especially with negative terms. It is helpful to explicitly label the terms and carefully track the common difference when moving along the sequence.

Final Answer:
The 14th term of the arithmetic progression is -19, which corresponds to option C.