The first term of a sequence is 9. Each term after the first is 4 times the preceding term. If...

1. TRANSLATE the problem information

• Given information:
  • First term is 9
  • Each term after the first is 4 times the preceding term
  • Need to find equation for w (the nth term) in terms of n

• What this tells us: This describes a geometric sequence where each term multiplies by the same factor (4) to get the next term.

2. INFER the sequence pattern

• Since each term is 4 times the previous term, I can build the sequence:
  • Term 1 (n = 1): \(\mathrm{w = 9}\)
  • Term 2 (n = 2): \(\mathrm{w = 9 \times 4 = 36}\)
  • Term 3 (n = 3): \(\mathrm{w = 9 \times 4 \times 4 = 9 \times 4^2}\)
  • Term 4 (n = 4): \(\mathrm{w = 9 \times 4^2 \times 4 = 9 \times 4^3}\)

• Pattern recognition: The nth term appears to be \(\mathrm{w = 9 \times 4^{(n-1)}}\)

3. INFER the general formula

• This matches the geometric sequence formula: \(\mathrm{a_n = a_1 \times r^{(n-1)}}\)
  • First term \(\mathrm{(a_1) = 9}\)
  • Common ratio \(\mathrm{(r) = 4}\)
  • So: \(\mathrm{w = 9 \times 4^{(n-1)}}\)

4. SIMPLIFY to verify the formula works

• Check n = 1: \(\mathrm{w = 9 \times 4^{(1-1)} = 9 \times 4^0 = 9 \times 1 = 9}\) ✓
• Check n = 2: \(\mathrm{w = 9 \times 4^{(2-1)} = 9 \times 4^1 = 9 \times 4 = 36}\) ✓

Looking at the answer choices, this matches choice D: \(\mathrm{w = 9(4^{(n-1)})}\)

Answer: D

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students might misinterpret "each term after the first is 4 times the preceding term" as meaning every term gets multiplied by 4, rather than understanding that 4 is the multiplier between consecutive terms.

This confusion leads them to think the sequence goes: 9, 9×4, (9×4)×4, etc., but they might incorrectly conclude the formula should have 4 as the base of the exponential. They might select Choice C (\(\mathrm{w = 9(4^n)}\)) because it "looks right" with 9 as the coefficient and 4 in the exponential, without checking that this gives \(\mathrm{w = 36}\) when \(\mathrm{n = 1}\).

Second Most Common Error:

Poor SIMPLIFY execution: Students might correctly identify the pattern as \(\mathrm{w = 9 \times 4^{(n-1)}}\) but make an error when checking their work. Specifically, they might forget that \(\mathrm{4^0 = 1}\), so when they check \(\mathrm{n = 1}\), they get confused about whether their formula gives the right first term.

This uncertainty might cause them to second-guess their correct reasoning and switch to an incorrect answer, or simply guess among the remaining choices.

The Bottom Line:

This problem tests whether students can recognize the structure of a geometric sequence and correctly translate the verbal description into mathematical notation. The key insight is understanding that the exponent must be \(\mathrm{(n-1)}\) to make the first term work correctly when the exponential base is the common ratio.