What is the sum of the first 13 terms of an arithmetic progression if the first term is −10 and the 13th (last given) term is 26?

Introduction / Context:
This question asks for the sum of a finite arithmetic progression (A.P.) when the first term and the last term of the sequence are known. The number of terms is 13. This is a direct application of the sum formula for an A.P.

Given Data / Assumptions:
The first term a is −10.The last (13th) term l is 26.The number of terms n is 13.We must find S13, the sum of the first 13 terms.

Concept / Approach:
For an arithmetic progression with n terms, first term a and last term l, the sum Sn is given by Sn = n * (a + l) / 2. Here we already know n, a and l, so there is no need to calculate the common difference. We simply substitute into the formula.

Step-by-Step Solution:
Step 1: Recall the sum formula Sn = n * (a + l) / 2.Step 2: Substitute n = 13, a = −10 and l = 26.Step 3: Compute a + l = −10 + 26 = 16.Step 4: Compute S13 = 13 * 16 / 2.Step 5: Simplify 16 / 2 = 8, so S13 = 13 * 8.Step 6: Multiply 13 by 8 to get S13 = 104.

Verification / Alternative check:
We can also confirm that 26 is indeed the 13th term. The common difference d can be found from the relation l = a + (n − 1)d: 26 = −10 + 12d. Then 26 + 10 = 36, so 12d = 36 and d = 3. The sequence is −10, −7, −4, −1, 2, 5, 8, 11, 14, 17, 20, 23, 26. Summing these manually gives the same result: pairs (−10, 26), (−7, 23), (−4, 20), (−1, 17), (2, 14), (5, 11) each add to 16 (six pairs), and the middle term 8. So total is 6 × 16 + 8 = 96 + 8 = 104.

Why Other Options Are Wrong:
Values like 140 or 98 could arise from miscalculating a + l or forgetting to divide by 2 in the sum formula. The value 84 would correspond to a shorter or different sequence. None of these obey Sn = n * (a + l) / 2 with the given values.

Common Pitfalls:
Some learners accidentally use (n − 1) instead of n in the sum formula or mix up a and l. Others forget that the formula already accounts for all n terms and attempt to add an extra term. Carefully substituting the values and simplifying step by step prevents such errors.

Final Answer:
The sum of the first 13 terms of the A.P. is 104.