Question:In an arithmetic sequence, the 3rd term is 14 and the 7th term is 30. Which equation gives the nth...
GMAT Algebra : (Alg) Questions
In an arithmetic sequence, the 3rd term is \(14\) and the 7th term is \(30\). Which equation gives the nth term of this sequence?
1. TRANSLATE the problem information
- Given information:
- \(\mathrm{a_3 = 14}\) (the 3rd term is 14)
- \(\mathrm{a_7 = 30}\) (the 7th term is 30)
- Need: the equation for \(\mathrm{a_n}\) (the general term)
2. INFER the solution strategy
- Since we have two terms from an arithmetic sequence, we can find the common difference d first
- Once we have d, we can work backwards to find \(\mathrm{a_1}\)
- Then construct the general formula \(\mathrm{a_n = a_1 + (n-1)d}\)
3. Find the common difference d
- In arithmetic sequences, d = (later term - earlier term)/(difference in positions)
- \(\mathrm{d = \frac{a_7 - a_3}{7 - 3}}\)
\(\mathrm{= \frac{30 - 14}{4}}\)
\(\mathrm{= \frac{16}{4}}\)
\(\mathrm{= 4}\)
4. Find the first term \(\mathrm{a_1}\)
- Using \(\mathrm{a_3 = 14}\) and the formula \(\mathrm{a_3 = a_1 + (3-1)d}\):
- \(\mathrm{14 = a_1 + 2(4)}\)
\(\mathrm{= a_1 + 8}\) - Therefore \(\mathrm{a_1 = 6}\)
5. SIMPLIFY to get the final formula
- \(\mathrm{a_n = a_1 + (n-1)d}\)
\(\mathrm{= 6 + (n-1)4}\) - \(\mathrm{a_n = 6 + 4n - 4}\)
\(\mathrm{= 4n + 2}\)
6. Verify with the given terms
- \(\mathrm{a_3 = 4(3) + 2 = 14}\) ✓
- \(\mathrm{a_7 = 4(7) + 2 = 30}\) ✓
Answer: D (\(\mathrm{a_n = 4n + 2}\))
Why Students Usually Falter on This Problem
Most Common Error Path:
Weak INFER skill: Students try to use the given terms directly without recognizing they need to find d first. They might attempt to create equations like "\(\mathrm{14 = 3a_1}\)" or try to guess relationships between 14, 30, and n. This approach leads to confusion and guessing rather than systematic solution.
Second Most Common Error:
Poor TRANSLATE reasoning: Students confuse the position numbers with coefficients or mix up which term corresponds to which value. For example, they might think "since \(\mathrm{a_3 = 14}\), then \(\mathrm{a_n = 14n}\)" or misread the relationship between term position and term value. This may lead them to select Choice A (\(\mathrm{a_n = 30n + 14}\)) by incorrectly combining the given numbers.
The Bottom Line:
This problem requires understanding that you can't directly jump to the final formula - you must work systematically through finding the common difference first, then the first term, then simplify to match the answer format.