The first few terms of an arithmetic sequence are shown in the table below.na_n211419627Which of the following gives the nth...

1. TRANSLATE the problem information

• Given information:
  • Three terms of an arithmetic sequence: \(\mathrm{a_2 = 11}\), \(\mathrm{a_4 = 19}\), \(\mathrm{a_6 = 27}\)
  • Need to find the general formula \(\mathrm{a_n}\)
• What this tells us: We have every other term, not consecutive terms

2. INFER the approach for finding common difference

• Since we have terms at positions 2, 4, and 6 (not consecutive), we need to be careful calculating d
• Between \(\mathrm{a_2}\) and \(\mathrm{a_4}\), there are 2 steps in the sequence
• So: \(\mathrm{a_4 = a_2 + 2d}\)

3. Calculate the common difference

• Set up: \(\mathrm{19 = 11 + 2d}\)
• Solve: \(\mathrm{2d = 8}\), so \(\mathrm{d = 4}\)

4. INFER how to find the first term

• Use the relationship: \(\mathrm{a_2 = a_1 + d}\)
• Substitute: \(\mathrm{11 = a_1 + 4}\)
• Therefore: \(\mathrm{a_1 = 7}\)

5. SIMPLIFY to get the final formula

• Start with: \(\mathrm{a_n = a_1 + (n-1)d}\)
• Substitute values: \(\mathrm{a_n = 7 + (n-1)(4)}\)
• Expand: \(\mathrm{a_n = 7 + 4n - 4}\)
• Simplify: \(\mathrm{a_n = 4n + 3}\)

6. INFER the need for verification

• Check all given values with our formula:
  • \(\mathrm{a_2 = 4(2) + 3 = 11}\) ✓
  • \(\mathrm{a_4 = 4(4) + 3 = 19}\) ✓
  • \(\mathrm{a_6 = 4(6) + 3 = 27}\) ✓

Answer: C

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students incorrectly calculate the common difference as \(\mathrm{d = a_4 - a_2 = 19 - 11 = 8}\), not realizing this spans two positions in the sequence.

Using \(\mathrm{d = 8}\), they would find \(\mathrm{a_1 = 11 - 8 = 3}\), leading to \(\mathrm{a_n = 3 + (n-1)(8) = 8n - 5}\). This formula doesn't match any answer choice, causing confusion and potentially random guessing.

Second Most Common Error:

Poor INFER reasoning about verification: Students find a formula that works for \(\mathrm{a_2 = 11}\) and immediately select that answer without checking the other given values. Since all four answer choices correctly give \(\mathrm{a_2 = 11}\), this incomplete verification could lead to selecting any of the wrong choices (A, B, or D).

The Bottom Line:

This problem tests whether students understand how to work with non-consecutive terms in arithmetic sequences and emphasizes the importance of complete verification with multiple data points.