In an arithmetic progression, the 18th term is 29 and the 29th term is 18. What is the value of the 49th term of this progression?

Introduction / Context:
This question uses standard properties of arithmetic progressions (APs). In an AP, the difference between consecutive terms is constant and is called the common difference. If we know any two terms and their positions in the sequence, we can find both the first term and the common difference. Once these are known, any other term can be computed using the general term formula. Here we are given the 18th and 29th terms and asked to find the 49th term.

Given Data / Assumptions:
- The sequence is an arithmetic progression.
- Let a be the first term and d be the common difference.
- The 18th term is 29, so a + 17d = 29.
- The 29th term is 18, so a + 28d = 18.
- We want the 49th term, which is a + 48d.

Concept / Approach:
We use the nth term formula for an arithmetic progression: T_n = a + (n - 1)d. By plugging in n = 18 and n = 29, we obtain two linear equations in a and d. Solving this system gives us the values of a and d. Then we substitute these into T_49 = a + 48d to get the required term. This method is general and works for any pair of known terms in an AP.

Step-by-Step Solution:
Step 1: Use the 18th term: T_18 = a + 17d = 29. Step 2: Use the 29th term: T_29 = a + 28d = 18. Step 3: Subtract the first equation from the second: (a + 28d) - (a + 17d) = 18 - 29. Step 4: This simplifies to 11d = -11, so d = -1. Step 5: Substitute d = -1 back into a + 17d = 29 to find a: a + 17(-1) = 29, so a - 17 = 29 and a = 46. Step 6: The 49th term is T_49 = a + 48d = 46 + 48(-1) = 46 - 48 = -2.

Verification / Alternative check:
We can check the given terms using a = 46 and d = -1. The 18th term is a + 17d = 46 + 17(-1) = 46 - 17 = 29, which matches the given value. The 29th term is a + 28d = 46 + 28(-1) = 46 - 28 = 18, which also matches. This confirms that the parameters of the AP are correct and that T_49 = -2 is reliable.

Why Other Options Are Wrong:
Options -3, 0, 1: None of these values result from the correct substitution into T_49. They correspond to incorrect values of d or miscalculations of the term index.
Option None of these: Incorrect because -2 is one of the listed options and is the correct 49th term.

Common Pitfalls:
A typical mistake is to mix up the (n - 1) factor and accidentally use n instead, which shifts the index by one and produces wrong terms. Another error is in subtracting the equations: if the signs are mishandled, a wrong common difference is obtained. Being careful with algebraic manipulation and using the standard T_n formula correctly avoids these problems.

Final Answer:
The 49th term of the arithmetic progression is -2.