In an arithmetic progression, the 1st term is -10 and the 3rd term is -4. What is the 12th term of this progression?

Introduction / Context:
This question deals with an arithmetic progression (A.P.), where each term increases by a constant difference. You are given the 1st and 3rd terms of the progression and asked to find the 12th term. The problem tests your understanding of how to use the general term formula of an arithmetic progression and how to compute the common difference from two known terms.

Given Data / Assumptions:

- First term a_1 is -10. - Third term a_3 is -4. - The progression has a constant common difference d. - We need to determine the 12th term a_12.

Concept / Approach:
The n-th term T_n of an arithmetic progression is given by T_n = a_1 + (n - 1)d. Knowing the first term and the third term allows us to compute the common difference. Once we have the common difference, we can directly calculate any term, including the 12th, by substituting n = 12 into the general term formula. This is a straightforward application of the definition of an arithmetic progression.

Step-by-Step Solution:
Step 1: Use the general term formula for T_3: T_3 = a_1 + (3 - 1)d = a_1 + 2d. Step 2: Substitute the known values: a_1 = -10 and T_3 = -4, so -10 + 2d = -4. Step 3: Solve for d: 2d = -4 + 10 = 6, so d = 3. Step 4: Now use the formula for the 12th term: T_12 = a_1 + (12 - 1)d = a_1 + 11d. Step 5: Substitute a_1 = -10 and d = 3: T_12 = -10 + 11 * 3. Step 6: Compute 11 * 3 = 33. Step 7: So T_12 = -10 + 33 = 23. Step 8: Therefore, the 12th term of the progression is 23.

Verification / Alternative check:
We can list some terms of the sequence to verify. Starting with -10 and adding 3 each time, we get: -10, -7, -4, -1, 2, 5, 8, 11, 14, 17, 20, 23, ... The 1st term is -10, the 3rd term is -4, as given. Counting to the 12th term, we indeed reach 23. This confirms that our calculated common difference and resulting 12th term are correct.

Why Other Options Are Wrong:
Options 20, 17 and 26 correspond to incorrect calculations, such as using the wrong number of steps (for example, 10d instead of 11d) or miscomputing the common difference. Only 23 is consistent with both the given terms and the constant step size of 3 between consecutive terms.

Common Pitfalls:
Some learners may incorrectly use d = (T_3 - a_1) / 3 instead of dividing by (3 - 1), leading to the wrong common difference. Others miscount when finding the 12th term and use 10 or 12 multipliers of d incorrectly. Remember that T_n uses (n - 1)d, and always double check the subtraction when computing the common difference.

Final Answer:
The 12th term of the arithmetic progression is 23, which corresponds to option D.