In an arithmetic progression, the third term is −13 and the sixth term is −4. Using properties of arithmetic progressions, what is the sum of the first 12 terms of this sequence?

Introduction / Context:
This problem tests core arithmetic progression concepts, including finding the first term, common difference and then using the formula for the sum of a finite number of terms. It requires manipulation of two term values to extract progression parameters.

Given Data / Assumptions:

• Third term T₃ = −13
• Sixth term T₆ = −4
• We consider a standard arithmetic progression with first term a and common difference d
• We need the sum of the first 12 terms S₁₂

Concept / Approach:
In an arithmetic progression, the nth term is Tₙ = a + (n − 1)d. We use the given T₃ and T₆ to form two equations in a and d. Once a and d are known, we use Sₙ = n/2 [2a + (n − 1)d] to compute the sum of the first 12 terms.

Step-by-Step Solution:
T₃ = a + 2d = −13 T₆ = a + 5d = −4 Subtract the first from the second: (a + 5d) − (a + 2d) = −4 − (−13) 3d = 9, so d = 3 Substitute d into a + 2d = −13 a + 2*3 = −13 gives a + 6 = −13 a = −19 Now use sum formula Sₙ = n/2 [2a + (n − 1)d] For n = 12: S₁₂ = 12/2 [2(−19) + 11*3] S₁₂ = 6[−38 + 33] = 6(−5) = −30

Verification / Alternative check:
We can list a few terms: a = −19, d = 3. So terms are −19, −16, −13, −10, −7, −4, and so on. The 3rd term is −13 and the 6th term is −4, matching the data. Pairing terms T₁ with T₁₂, T₂ with T₁₁, and so forth also yields a consistent sum of −30 when calculated carefully.

Why Other Options Are Wrong:
67 and 45 are positive and ignore the heavily negative early terms. −48 has too large a magnitude and does not match the formula based calculation. 30 has the wrong sign because the progression remains largely negative over the first 12 terms.

Common Pitfalls:
Errors often arise when forming equations for T₃ and T₆, particularly with the (n − 1)d term. Another pitfall is mixing up signs while substituting a and d into the sum formula. Students sometimes use n in place of (n − 1) inside the bracket which changes the final answer significantly.

Final Answer:
Therefore, the sum of the first 12 terms of the arithmetic progression is −30.