The first term of an arithmetic progression is -10 and the thirteenth term is 26. What is the sum of the first 13 terms of this arithmetic progression?

Introduction / Context:
This problem checks your knowledge of arithmetic progressions and in particular the formula for the sum of the first n terms of an arithmetic progression. You are given the first term and the 13th term and asked to find the sum of the first 13 terms. This is a direct application of the standard sum formula and is a typical question in aptitude tests to see whether you remember formulas and can apply them correctly.

Given Data / Assumptions:

• First term of the arithmetic progression, a1 = -10.
• Thirteenth term of the arithmetic progression, a13 = 26.
• Number of terms to be summed, n = 13.
• The sequence is a standard arithmetic progression with constant common difference.

Concept / Approach:
For any arithmetic progression, the sum of the first n terms is given by the formula S_n = n * (a1 + a_n) / 2, where a1 is the first term and a_n is the nth term. In this question, we are provided directly with a1 and a13, so we do not even need to calculate the common difference. The question reduces to substituting n = 13, a1 = -10, and a13 = 26 into the formula and simplifying carefully.

Step-by-Step Solution:
We have n = 13, a1 = -10, and a13 = 26. The sum of the first n terms of an arithmetic progression is S_n = n * (a1 + a_n) / 2. Here S_13 = 13 * (a1 + a13) / 2. Substitute the values: S_13 = 13 * (-10 + 26) / 2. Compute the sum in the numerator: -10 + 26 = 16. So S_13 = 13 * 16 / 2. Divide 16 by 2 first: 16 / 2 = 8. Now multiply: S_13 = 13 * 8 = 104. Therefore, the sum of the first 13 terms of the arithmetic progression is 104.

Verification / Alternative Check:
As a quick verification, we can find the common difference and a few terms. The common difference d is (a13 - a1) / 12 = (26 - (-10)) / 12 = 36 / 12 = 3. So the sequence is -10, -7, -4, -1, 2, 5, 8, 11, 14, 17, 20, 23, 26. We can sum a few terms in pairs from the ends: (-10 + 26) = 16, (-7 + 23) = 16, (-4 + 20) = 16, (-1 + 17) = 16, (2 + 14) = 16, (5 + 11) = 16 and the middle term is 8. There are 6 such pairs each summing to 16 and one middle term. So total sum is 6 * 16 + 8 = 96 + 8 = 104, which matches the formula calculation.

Why Other Options Are Wrong:

• Option 140: This would require either a wrong value for the number of terms or a miscalculation in the formula and does not match the correct sum.
• Option 84: This is smaller than the correct value and usually appears if someone mistakenly uses 12 instead of 13 for the number of terms.
• Option 98: This could arise from an arithmetic slip when multiplying or dividing, but it does not satisfy the exact sum when checked.
• Option 156: This would correspond to a much larger average term and does not fit the actual sequence values.

Common Pitfalls:
Some students forget the correct sum formula and instead try to add many terms manually, which is time consuming and prone to error. Others may confuse the formula for the sum with the formula for the nth term, or mistakenly use n - 1 instead of n in the sum formula. Arithmetic errors, especially when dealing with negative first terms, are also common. It is important to first compute a1 + a_n correctly, then multiply by n, and only then divide by 2 to avoid mistakes.

Final Answer:
The sum of the first 13 terms of the arithmetic progression is 104.