In an arithmetic progression the first term is 23, the last term is 42, and there are 14 terms in total. What is the sum of all the terms in this series?

Introduction / Context:
This question is about arithmetic progressions and asks you to compute the sum of the terms. You are given the first term, the last term, and the total number of terms. Even if the common difference is not an integer, the sum formula still applies.

Given Data / Assumptions:
- First term a = 23.
- Last term l = 42.
- Number of terms n = 14.
- The series is an arithmetic progression where consecutive terms differ by a constant amount.

Concept / Approach:
The sum S of an arithmetic progression with n terms, first term a, and last term l is given by S = n * (a + l) / 2. It is not necessary to know the common difference to use this formula. We simply substitute the given values and compute.

Step-by-Step Solution:
Step 1: Identify the known values: a = 23, l = 42, n = 14.Step 2: Use the sum formula S = n * (a + l) / 2.Step 3: Compute the sum inside the parentheses: a + l = 23 + 42 = 65.Step 4: Multiply by the number of terms: n * (a + l) = 14 * 65.Step 5: Calculate 14 * 65 = 910.Step 6: Divide by 2 to get S = 910 / 2 = 455.Step 7: Therefore, the sum of the 14 term arithmetic progression is 455.

Verification / Alternative check:
We can check consistency by calculating the common difference d. Using l = a + (n - 1) * d gives 42 = 23 + 13d. So 19 = 13d and d = 19 / 13. The progression is 23, 23 + 19 / 13, and so on up to 42. Even though the difference is fractional, the sum formula S = n * (a + l) / 2 remains valid for any arithmetic progression, and we have already applied it correctly.

Why Other Options Are Wrong:
Options 364, 420, 504, and 315 result from incorrect calculations such as dividing by 14 instead of 2, adding terms incorrectly, or misreading the formula. None of these values equals n * (a + l) / 2 when n = 14, a = 23, and l = 42. Only 455 matches the correct result from the standard formula.

Common Pitfalls:
Some learners mistakenly use S = (a + l) / 2 without multiplying by n, missing the crucial factor of the number of terms. Others try to generate all terms explicitly and then add them, which is time consuming and prone to arithmetic mistakes. Remembering and trusting the formula S = n * (a + l) / 2 is the most efficient method.

Final Answer:
The sum of all 14 terms in the arithmetic progression is 455.