In an arithmetic sequence, the 8^(th) term is 23 and the 12^(th) term is 17. Which equation gives the n^(th)...

1. TRANSLATE the problem information

• Given information:
  • 8th term: \(\mathrm{a_8 = 23}\)
  • 12th term: \(\mathrm{a_{12} = 17}\)
  • Need: formula for nth term

2. INFER the solution strategy

• Since we have two terms from an arithmetic sequence, we can find the common difference first
• The common difference is the slope between any two terms: \(\mathrm{d = \frac{change\,in\,term\,value}{change\,in\,position}}\)

3. SIMPLIFY to find the common difference

\(\mathrm{d = \frac{a_{12} - a_8}{12 - 8}}\)

\(\mathrm{= \frac{17 - 23}{4}}\)

\(\mathrm{= \frac{-6}{4}}\)

\(\mathrm{= -\frac{3}{2}}\)

4. INFER how to find the first term

• Use the arithmetic sequence formula with one known term
• Since \(\mathrm{a_8 = a_1 + 7d}\), we can solve for \(\mathrm{a_1}\)

5. SIMPLIFY to find the first term

\(\mathrm{23 = a_1 + 7(-\frac{3}{2})}\)

\(\mathrm{23 = a_1 - \frac{21}{2}}\)

\(\mathrm{a_1 = 23 + \frac{21}{2}}\)

\(\mathrm{= \frac{46}{2} + \frac{21}{2}}\)

\(\mathrm{= \frac{67}{2}}\)

6. SIMPLIFY to convert to slope-intercept form

\(\mathrm{a_n = \frac{67}{2} + (n-1)(-\frac{3}{2})}\)

\(\mathrm{a_n = \frac{67}{2} - \frac{3}{2}n + \frac{3}{2}}\)

\(\mathrm{a_n = (\frac{67}{2} + \frac{3}{2}) - \frac{3}{2}n}\)

\(\mathrm{a_n = \frac{70}{2} - \frac{3}{2}n}\)

\(\mathrm{a_n = 35 - \frac{3}{2}n}\)

\(\mathrm{a_n = -\frac{3}{2}n + 35}\)

Answer: A

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students make arithmetic errors when combining fractions, especially when finding \(\mathrm{a_1 = \frac{67}{2} + \frac{3}{2} = 35}\). They might incorrectly calculate this as 32, 29, or 38.

If they get \(\mathrm{a_1 = \frac{64}{2} = 32}\), this leads them to select Choice C \(\mathrm{(a_n = -\frac{3}{2}n + 32)}\)

If they get \(\mathrm{a_1 = \frac{58}{2} = 29}\), this leads them to select Choice D \(\mathrm{(a_n = -\frac{3}{2}n + 29)}\)

If they get \(\mathrm{a_1 = \frac{76}{2} = 38}\), this leads them to select Choice E \(\mathrm{(a_n = -\frac{3}{2}n + 38)}\)

Second Most Common Error:

Poor INFER reasoning: Students incorrectly calculate the common difference by mixing up which term to subtract from which, getting \(\mathrm{d = \frac{23-17}{4} = \frac{6}{4} = \frac{3}{2}}\) instead of \(\mathrm{d = -\frac{3}{2}}\).

This incorrect positive slope leads them to select Choice B \(\mathrm{(a_n = -\frac{2}{3}n + 35)}\) after making additional errors, or causes confusion and guessing.

The Bottom Line:

This problem tests your ability to work systematically with arithmetic sequences and carefully handle fraction arithmetic. The key insight is recognizing that an arithmetic sequence is really just a linear function, so finding the common difference (slope) and first term lets you write it in slope-intercept form.