In an arithmetic sequence, the first term is 17 and the second term is 25. Which equation defines the nth...
GMAT Algebra : (Alg) Questions
In an arithmetic sequence, the first term is 17 and the second term is 25. Which equation defines the nth term \(\mathrm{a_n}\)?
- \(\mathrm{a_n = \frac{1}{8}n + 9}\)
- \(\mathrm{a_n = n + 16}\)
- \(\mathrm{a_n = 8n + 17}\)
- \(\mathrm{a_n = 8n + 9}\)
1. TRANSLATE the problem information
- Given information:
- First term: \(\mathrm{a_1 = 17}\)
- Second term: \(\mathrm{a_2 = 25}\)
- Need to find: equation for \(\mathrm{a_n}\) (the nth term)
2. INFER the approach
- Since this is an arithmetic sequence, consecutive terms have a constant difference
- Strategy: Find the common difference first, then use the general formula \(\mathrm{a_n = a_1 + (n-1)d}\)
3. Calculate the common difference
- \(\mathrm{d = a_2 - a_1 = 25 - 17 = 8}\)
4. Apply the arithmetic sequence formula and SIMPLIFY
- \(\mathrm{a_n = a_1 + (n-1)d}\)
- \(\mathrm{a_n = 17 + (n-1) \times 8}\)
- \(\mathrm{a_n = 17 + 8n - 8}\)
- \(\mathrm{a_n = 8n + 9}\)
5. Verify the answer
- For \(\mathrm{n = 1: a_1 = 8(1) + 9 = 17}\) ✓
- For \(\mathrm{n = 2: a_2 = 8(2) + 9 = 25}\) ✓
Answer: D
Why Students Usually Falter on This Problem
Most Common Error Path:
Weak SIMPLIFY skill: Students make algebraic mistakes when expanding \(\mathrm{17 + (n-1) \times 8}\), particularly forgetting to distribute the 8 to both n and -1, or making sign errors when combining like terms.
Common mistake: \(\mathrm{17 + (n-1) \times 8}\) becomes \(\mathrm{17 + 8n}\) (forgetting to multiply \(\mathrm{8 \times (-1)}\)), leading to \(\mathrm{a_n = 8n + 17}\).
This may lead them to select Choice C (\(\mathrm{a_n = 8n + 17}\)).
Second Most Common Error:
Conceptual confusion about arithmetic sequence formula: Students remember a formula but use \(\mathrm{a_n = a_1 + nd}\) instead of the correct \(\mathrm{a_n = a_1 + (n-1)d}\).
Using the wrong formula: \(\mathrm{a_n = 17 + n \times 8 = 8n + 17}\), again leading to the same incorrect result.
This may lead them to select Choice C (\(\mathrm{a_n = 8n + 17}\)).
The Bottom Line:
This problem tests whether students can correctly apply and manipulate the arithmetic sequence formula. The key insight is remembering that the general term uses (n-1) as the multiplier for the common difference, not just n.