In an arithmetic progression, the sum of the first 13 terms is required. The 5th term of the A.P. is 1 and the 8th term is −17. What is the sum of the first 13 terms?

Introduction / Context:
This problem deals with an arithmetic progression (A.P.) where two specific terms are known and we must find the sum of the first 13 terms. It combines the formula for the n-th term of an A.P. with the formula for the sum of the first n terms, requiring you to solve for both the first term and common difference.

Given Data / Assumptions:
Let the first term be a and common difference be d.The 5th term T5 is 1.The 8th term T8 is −17.We must find the sum S13 of the first 13 terms.

Concept / Approach:
The n-th term of an A.P. is Tn = a + (n − 1)d. The sum Sn of the first n terms is Sn = n * (first term + last term) / 2, or Sn = n/2 * [2a + (n − 1)d]. First, we use T5 and T8 to solve for a and d. Then we find the 13th term T13 and finally use the sum formula.

Step-by-Step Solution:
Step 1: Write T5 and T8 using Tn = a + (n − 1)d.Step 2: T5 = a + 4d = 1.Step 3: T8 = a + 7d = −17.Step 4: Subtract the equation for T5 from the equation for T8: (a + 7d) − (a + 4d) = −17 − 1.Step 5: This gives 3d = −18, so d = −6.Step 6: Substitute d = −6 into a + 4d = 1: a + 4(−6) = 1, so a − 24 = 1 and a = 25.Step 7: Now find the 13th term T13 = a + 12d = 25 + 12(−6) = 25 − 72 = −47.Step 8: Use the sum formula S13 = 13 * (first term + last term) / 2.Step 9: S13 = 13 * (25 + (−47)) / 2 = 13 * (−22) / 2 = 13 * (−11) = −143.

Verification / Alternative check:
We can check some terms to ensure the pattern is correct. With a = 25 and d = −6, the sequence begins: 25, 19, 13, 7, 1, −5, −11, −17, ... T5 is 1 and T8 is −17, matching the given values. Continuing this pattern down to T13 and summing the first 13 terms using the formula confirms S13 = −143.

Why Other Options Are Wrong:
Options like −140 or 61 result from miscalculations, such as using the wrong number of terms or incorrect values of a or d. A positive sum like 166 contradicts the fact that many later terms in the progression are negative and of larger magnitude, so the overall sum must be negative.

Common Pitfalls:
Students often forget that Tn = a + (n − 1)d, not a + nd. Another frequent error is to mis-handle the negative common difference or to use the wrong formula for the sum. Careful algebra and explicit substitution into the formulas help avoid these issues.

Final Answer:
The sum of the first 13 terms of the A.P. is −143.