An arithmetic sequence has nth term \(\mathrm{T(n) = pn + q}\), where p and q are constants. The second term...

1. TRANSLATE the problem information

• Given information:
  • Arithmetic sequence with nth term: \(\mathrm{T(n) = pn + q}\)
  • Second term: \(\mathrm{T(2) = 7}\)
  • Eighth term: \(\mathrm{T(8) = 43}\)
  • Need to find: \(\mathrm{p - q}\)

2. TRANSLATE the sequence information into equations

• Substitute the known values into the general formula:
  • \(\mathrm{T(2) = 2p + q = 7}\)
  • \(\mathrm{T(8) = 8p + q = 43}\)

• This gives us a system of two linear equations with two unknowns.

3. SIMPLIFY to solve for p

• Subtract the first equation from the second to eliminate q:
  \(\mathrm{(8p + q) - (2p + q) = 43 - 7}\)
  \(\mathrm{6p = 36}\)
  \(\mathrm{p = 6}\)

4. SIMPLIFY to solve for q

• Substitute \(\mathrm{p = 6}\) back into the first equation:
  \(\mathrm{2(6) + q = 7}\)
  \(\mathrm{12 + q = 7}\)
  \(\mathrm{q = -5}\)

5. SIMPLIFY to find p - q

• Calculate the final answer:
  \(\mathrm{p - q = 6 - (-5) = 6 + 5 = 11}\)

Answer: (D) 11

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY skill: Sign error when calculating p - q with negative q

Students correctly find \(\mathrm{p = 6}\) and \(\mathrm{q = -5}\), but then calculate \(\mathrm{p - q = 6 - 5 = 1}\) instead of recognizing that subtracting a negative number means adding: \(\mathrm{6 - (-5) = 6 + 5 = 11}\).

This may lead them to select an incorrect answer or become confused when 1 doesn't match any answer choice, leading to guessing.

Second Most Common Error:

Poor TRANSLATE reasoning: Setting up incorrect equations from the sequence information

Students might confuse the arithmetic sequence formula or incorrectly substitute values, such as writing \(\mathrm{T(n) = p + qn}\) instead of \(\mathrm{T(n) = pn + q}\), or mixing up which term corresponds to which position.

This leads to completely different values for p and q, causing them to select the wrong answer choice.

The Bottom Line:

This problem tests both systematic equation-solving skills and careful attention to sign rules. The key insight is recognizing that arithmetic sequences in the form \(\mathrm{T(n) = pn + q}\) create a simple system of linear equations when you're given specific terms.