In an arithmetic progression, the 3rd term is -9 and the 7th term is 11. What is the 15th term of this progression?

Introduction / Context:
This arithmetic progression question gives two non consecutive terms and asks for another term further along the sequence. Specifically, you know the 3rd and 7th terms and must find the 15th term. Since an arithmetic progression has a constant common difference, the two known terms are enough to determine both the first term and the common difference, after which finding any term is straightforward. The question reinforces the use of the general term formula and solving simple simultaneous equations.

Given Data / Assumptions:

- Let the first term be a. - Let the common difference be d. - The 3rd term T_3 is -9. - The 7th term T_7 is 11. - We must find the 15th term T_15.

Concept / Approach:
The formula for the n-th term of an arithmetic progression is T_n = a + (n - 1)d. Using T_3 and T_7, we can write two equations involving a and d. Subtracting these equations removes a and lets us solve for d, the common difference. Once we know d, we can substitute back to find a. With both a and d known, computing T_15 using the same general formula becomes straightforward. This structured approach works for all similar A.P. problems.

Step-by-Step Solution:
Step 1: Write T_3 using the general formula: T_3 = a + (3 - 1)d = a + 2d. Step 2: Given T_3 = -9, we get a + 2d = -9. Call this Equation (1). Step 3: Write T_7 using the general formula: T_7 = a + (7 - 1)d = a + 6d. Step 4: Given T_7 = 11, we get a + 6d = 11. Call this Equation (2). Step 5: Subtract Equation (1) from Equation (2): (a + 6d) - (a + 2d) = 11 - (-9). Step 6: This simplifies to 4d = 20. Step 7: Solve for d: d = 20 / 4 = 5. Step 8: Substitute d = 5 into Equation (1): a + 2 * 5 = -9, so a + 10 = -9. Step 9: Thus a = -9 - 10 = -19. Step 10: Now find T_15 using T_15 = a + (15 - 1)d = a + 14d. Step 11: Substitute a = -19 and d = 5: T_15 = -19 + 14 * 5. Step 12: Compute 14 * 5 = 70. Therefore T_15 = -19 + 70 = 51.

Verification / Alternative check:
List several terms of the progression to verify. With a = -19 and d = 5, the sequence begins -19, -14, -9, -4, 1, 6, 11, 16, 21, 26, 31, 36, 41, 46, 51, ... The 3rd term is -9 and the 7th term is 11, matching the given information. The 15th term in this list is indeed 51, confirming that the computation is correct.

Why Other Options Are Wrong:
Values 28, 87 and 17 do not match the term obtained from a consistent arithmetic progression with T_3 = -9 and T_7 = 11. These incorrect answers typically come from an incorrect common difference or misusing the term index in the formula (for example, using 15d instead of 14d). Only 51 fits both the data given and the constant difference of 5 between consecutive terms.

Common Pitfalls:
Students often mis-handle the subtraction step when solving for d, particularly with negative numbers, leading to an incorrect common difference. Another frequent error is to confuse T_n = a + (n - 1)d with T_n = a + nd. Carefully writing down and checking each equation, especially the indices and signs, helps avoid such mistakes.

Final Answer:
The 15th term of the arithmetic progression is 51, which corresponds to option C.