The table below shows several values of x and their corresponding values of \(\mathrm{g(x)}\) for the exponential function g, where...

1. TRANSLATE the problem information

• Given information:
  • \(\mathrm{g(x)}\) is an exponential function with base \(\mathrm{b \gt 1}\)
  • Table showing: \(\mathrm{g(2) = b^2}\), \(\mathrm{g(3) = b^6}\), \(\mathrm{g(4) = b^{12}}\)
  • Need to find k where \(\mathrm{g(k) = b^{30}}\)

2. INFER the pattern in exponents

• Look at the exponents: 2, 6, 12
• Test relationship between x-value and exponent:
  • \(\mathrm{x = 2}\): exponent \(\mathrm{= 2 = 2(1) = 2(2-1)}\) ✓
  • \(\mathrm{x = 3}\): exponent \(\mathrm{= 6 = 3(2) = 3(3-1)}\) ✓
  • \(\mathrm{x = 4}\): exponent \(\mathrm{= 12 = 4(3) = 4(4-1)}\) ✓
• Pattern discovered: \(\mathrm{g(x) = b^{x(x-1)}}\)

3. TRANSLATE the condition g(k) = b³⁰

• Using our pattern: \(\mathrm{b^{k(k-1)} = b^{30}}\)
• Since bases are equal: \(\mathrm{k(k-1) = 30}\)

4. SIMPLIFY the quadratic equation

• Expand: \(\mathrm{k^2 - k = 30}\)
• Standard form: \(\mathrm{k^2 - k - 30 = 0}\)
• Factor: Look for two numbers that multiply to -30 and add to -1
• Those numbers are -6 and +5: \(\mathrm{(k - 6)(k + 5) = 0}\)
• Solutions: \(\mathrm{k = 6}\) or \(\mathrm{k = -5}\)

5. APPLY CONSTRAINTS to select final answer

• Since the table shows positive x-values and we're looking for a value in the same context: \(\mathrm{k = 6}\)

Answer: B) 6

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students fail to recognize the \(\mathrm{x(x-1)}\) pattern and instead try to find a simple multiplicative or additive pattern between consecutive terms.

They might notice that \(\mathrm{6 = 3 \times 2}\) and \(\mathrm{12 = 2 \times 6}\), thinking there's a "multiply by the x-value" or "double the previous" pattern. This leads them to set up incorrect equations that don't yield any of the given answer choices, causing them to get stuck and guess randomly.

Second Most Common Error:

Inadequate SIMPLIFY execution: Students correctly identify the pattern and set up \(\mathrm{k(k-1) = 30}\), but make algebraic errors when solving the quadratic equation.

Common mistakes include incorrect factoring (like trying \(\mathrm{(k-5)(k+6) = 0}\)) or calculation errors when expanding \(\mathrm{k(k-1)}\). This may lead them to select Choice A (5) if they incorrectly factor or Choice C (7) if they make arithmetic mistakes.

The Bottom Line:

This problem requires pattern recognition skills that go beyond simple arithmetic sequences. Students must see the relationship between position and exponent, then execute multi-step algebraic manipulation correctly.