In an arithmetic progression, the first term is 7 and the ninth (last) term is 55. What is the sum of the first 9 terms of this arithmetic progression?

Introduction / Context:
This question checks your understanding of arithmetic progression, often abbreviated as AP. In an AP, each term differs from the previous one by a constant amount called the common difference. Here, you are given the first term, the ninth term, and asked to find the sum of the first nine terms. This is a basic but very important formula based topic in quantitative aptitude and mathematics.

Given Data / Assumptions:
- The sequence is an arithmetic progression.
- The first term a is 7.
- The ninth term a9 is 55.
- The number of terms n to be summed is 9.
- We are asked to find the sum of the first 9 terms, denoted by S9.

Concept / Approach:
For an arithmetic progression, the sum of the first n terms is given by the formula S_n = n * (first term + last term) / 2 when the first and nth term are known. In this problem, the first term and the ninth term are given directly, so we can apply the formula immediately without first computing the common difference. This makes the calculation straightforward and quick.

Step-by-Step Solution:
Step 1: Identify the known values: first term a = 7, ninth term a9 = 55, number of terms n = 9.Step 2: Use the sum formula for an arithmetic progression: S_n = n * (a + a_n) / 2.Step 3: Substitute n = 9, a = 7, and a_n = 55 into the formula to get S_9 = 9 * (7 + 55) / 2.Step 4: Compute the sum inside the brackets: 7 + 55 = 62.Step 5: The expression becomes S_9 = 9 * 62 / 2.Step 6: Divide 62 by 2 to simplify: 62 / 2 = 31.Step 7: Multiply 9 by 31 to get S_9 = 9 * 31 = 279.

Verification / Alternative check:
You can also compute the common difference d first. The formula for the nth term is a_n = a + (n - 1) * d. For n = 9, we have 55 = 7 + 8 * d, so 48 = 8d and d = 6. The sequence is then 7, 13, 19, 25, 31, 37, 43, 49, 55. Summing these manually: (7 + 55) + (13 + 49) + (19 + 43) + (25 + 37) + 31 = 62 + 62 + 62 + 62 + 31 = 4 * 62 + 31 = 248 + 31 = 279. This matches the result from the formula.

Why Other Options Are Wrong:
Values like 219, 231, 137, or 207 do not match the exact sum of the terms in this arithmetic progression. If you try to use them as S_9 in the reverse formula, you will not be able to express them as 9 times the average of 7 and 55. They mainly result from miscalculations, such as using the wrong number of terms or misapplying the formula.

Common Pitfalls:
Some learners mistakenly use the common difference incorrectly or apply the formula for an arithmetic series with the wrong values of first and last term. Others compute only the difference between first and last term or forget to divide by 2. Always remember that the sum formula involves both the first and nth terms and includes division by 2 to account for pairing of terms.

Final Answer:
The sum of the first 9 terms of the arithmetic progression is 279.