The first term of an arithmetic sequence is 3, and the common difference is 4. What is the value of...

1. INFER the problem type and strategy

• This is asking for a specific term in an arithmetic sequence
• Strategy: Use the arithmetic sequence formula since we have the first term and common difference
• Formula needed: \(\mathrm{a_n = a_1 + (n-1)d}\)

2. TRANSLATE the given information

• Given information:
  • First term: \(\mathrm{a_1 = 3}\)
  • Common difference: \(\mathrm{d = 4}\)
  • Term we want: \(\mathrm{n = 5}\) (the 5th term)

3. SIMPLIFY by substituting into the formula

• \(\mathrm{a_5 = a_1 + (n-1)d}\)
• \(\mathrm{a_5 = 3 + (5-1) \times 4}\)

4. SIMPLIFY the calculation step by step

• First: \(\mathrm{(5-1) = 4}\)
• Next: \(\mathrm{4 \times 4 = 16}\)
• Finally: \(\mathrm{3 + 16 = 19}\)

Answer: C) 19

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Student doesn't recognize this as an arithmetic sequence problem requiring the standard formula. Instead, they might try to manually count terms: "First term is 3, second is 3+4=7, third is 7+4=11, fourth is 11+4=15, fifth is 15+4=19." While this gives the correct answer, it's inefficient and prone to counting errors on longer sequences.

Second Most Common Error:

Poor SIMPLIFY execution: Student uses the correct formula but makes arithmetic mistakes. Common calculation errors include:

• Computing (5-1) as 6 instead of 4
• Getting \(\mathrm{4 \times 4 = 12}\) instead of 16
• Adding 3 + 16 incorrectly

These arithmetic slips could lead them to select Choice A (15) or Choice B (16).

The Bottom Line:

This problem tests whether students can quickly identify an arithmetic sequence scenario and efficiently apply the formula rather than manually counting terms. The key insight is recognizing that the arithmetic sequence formula provides a direct path to any term.