The first term of an arithmetic progression is -19 and the twelfth term is 36. What is the sum of the first 12 terms of this arithmetic progression?

Introduction / Context:
This is another straightforward arithmetic progression sum question. We are given the first term and the twelfth term and asked to find the sum of the first twelve terms. It reinforces the basic formula for the sum of an arithmetic series and is typical of quick scoring questions in many competitive exams.

Given Data / Assumptions:

• First term a1 = -19.
• Twelfth term a12 = 36.
• Number of terms n = 12.
• The sequence is an arithmetic progression with constant common difference.

Concept / Approach:
The sum S_n of the first n terms of an arithmetic progression is given by S_n = n * (a1 + a_n) / 2. The convenience in this question is that both the first term and the twelfth term are directly given, so we do not need to compute the common difference for the sum. By substituting n = 12, a1 = -19 and a12 = 36 into the formula, we can obtain the required sum quickly.

Step-by-Step Solution:
We have n = 12, a1 = -19, and a12 = 36. Use the sum formula S_n = n * (a1 + a_n) / 2. So S_12 = 12 * (a1 + a12) / 2. Substitute values: S_12 = 12 * (-19 + 36) / 2. Compute the sum inside the brackets: -19 + 36 = 17. So S_12 = 12 * 17 / 2. Simplify by dividing 12 by 2: 12 / 2 = 6. Now S_12 = 6 * 17 = 102. Therefore, the sum of the first 12 terms is 102.

Verification / Alternative Check:
As a quick check, we can find the common difference to see whether the numbers look reasonable. The common difference d is (a12 - a1) / 11 = (36 - (-19)) / 11 = 55 / 11 = 5. The sequence is then -19, -14, -9, -4, 1, 6, 11, 16, 21, 26, 31, 36. If we roughly average the first and last term, we get ( -19 + 36 ) / 2 = 17 / 2 = 8.5. Multiplying this average by 12 terms gives 8.5 * 12 = 102, which matches our formula result and confirms it is consistent.

Why Other Options Are Wrong:

• Option 192: This would require a much larger average term and is not consistent with the actual terms of the sequence.
• Option 230: This is significantly higher than the correct sum and cannot be obtained from the given values.
• Option 214: This is also too large and would imply an incorrect application of the sum formula or wrong arithmetic.
• Option 120: Slightly higher than the correct sum but still incorrect and likely arises from a miscalculation when multiplying or dividing.

Common Pitfalls:
Some learners mistakenly use the formula S_n = n * a1 instead of including the last term, which is incorrect for sums of arithmetic progressions. Others may accidentally use 11 as the number of terms because they misread the given twelfth term as implying eleven intervals rather than twelve entries. The safest approach is to clearly identify n, a1, and a_n, substitute carefully into the formula, and simplify step by step, especially when negative numbers are involved.

Final Answer:
The sum of the first 12 terms of the arithmetic progression is 102.