What is the sum of the first 12 terms of an arithmetic progression if the first term is 3 and the twelfth (last) term is 47?

Introduction / Context:
This question checks knowledge of arithmetic progressions and, in particular, the formula for the sum of the first n terms when the first term and the nth term are known. Arithmetic progressions appear frequently in quantitative aptitude and algebra topics.

Given Data / Assumptions:
- First term of the arithmetic progression, a = 3.
- Twelfth term is the last term mentioned, so l = 47.
- Number of terms n = 12.
- We need to calculate the sum S of the first 12 terms.

Concept / Approach:
The standard formula for the sum of the first n terms of an arithmetic progression is S = n / 2 * (first term + last term). This works whenever we know the first term a and the nth term l. We apply this formula directly using the given values.

Step-by-Step Solution:
Identify n = 12, a = 3, and l = 47. Use the sum formula S = n / 2 * (a + l). Compute a + l = 3 + 47 = 50. Compute n / 2 = 12 / 2 = 6. Therefore S = 6 * 50 = 300.

Verification / Alternative Check:
Another way is to note that the average of the first and last terms is (3 + 47) / 2 = 25. Since there are 12 terms, the sum is average × number of terms = 25 × 12 = 300. This matches the result obtained from the formula, verifying the correctness.

Why Other Options Are Wrong:
260 would correspond to a smaller average term or fewer terms; it does not match 25 × 12.
280 also does not equal 25 × 12 and indicates miscalculation, perhaps mixing up the number of terms.
220 is far too small and might come from incorrectly using half the number of terms or an incorrect last term.
240 is again inconsistent with both the correct formula and simple averaging.

Common Pitfalls:
Students sometimes confuse the formula for the nth term with the formula for the sum. Another frequent mistake is misidentifying n; for example, treating 47 as if it were the number of terms instead of the last term. Carefully reading the statement and clearly labelling a, l, and n prevents these errors.

Final Answer:
The sum of the first 12 terms of the arithmetic progression is 300.