The first and last terms of an arithmetic progression (A.P.) are 25 and 52 respectively. If the progression contains 12 terms in total, what is the sum of all the terms of this A.P.?

Introduction / Context:
This problem is based on sequences and series, specifically arithmetic progressions. You are given the first term, the last term, and the number of terms of an A.P., and asked to compute the total sum of all the terms. This is a standard formula-based question often seen in aptitude and competitive exams.

Given Data / Assumptions:

• First term a1 = 25.
• Last term a12 = 52.
• Number of terms n = 12.
• The sequence is arithmetic, meaning the difference between consecutive terms is constant.

Concept / Approach:
For an arithmetic progression with n terms, first term a1 and last term an, the sum S of the n terms is given by the formula: S = (n / 2) * (a1 + an). We do not even need the common difference if we know the first and last terms and the number of terms. Substituting the given values into this formula directly yields the required sum.

Step-by-Step Solution:
Step 1: Identify n, a1 and an. n = 12, a1 = 25, an = 52. Step 2: Use the sum formula for an arithmetic progression: S = (n / 2) * (a1 + an). Step 3: Substitute the values: S = (12 / 2) * (25 + 52). Step 4: Simplify: 12 / 2 = 6 and 25 + 52 = 77. Step 5: Multiply: S = 6 * 77 = 462. Step 6: Therefore, the sum of the 12 terms of the progression is 462.

Verification / Alternative check:
We can quickly check the reasonableness of the answer. The average of the first and last terms is (25 + 52) / 2 = 77 / 2 = 38.5. Since there are 12 terms, the sum should be 12 * 38.5 = 462. This matches the value obtained from the standard formula, confirming the correctness of the solution.

Why Other Options Are Wrong:
372, 252, 110 and 520 do not satisfy the arithmetic progression sum formula when the first term is 25, the last term is 52, and n = 12. They arise from either miscounting the number of terms, misusing the formula, or arithmetic errors.

Common Pitfalls:
One common error is to use the wrong formula, such as multiplying the first term by the number of terms without considering the last term. Another frequent mistake is to miscalculate the average or to forget to divide n by 2. Always recall the correct sum formula and check that you have substituted the correct values.

Final Answer:
The sum of all the terms of the A.P. is 462.