A ball is dropped from an initial height of 80 feet. After each time it hits the ground, it rebounds...

1. TRANSLATE the problem information

• Given information:
  • Initial drop height: 80 feet
  • Each bounce reaches 50% of previous height
  • \(\mathrm{h}\) = maximum height after nth bounce
  • Need equation for \(\mathrm{h}\) in terms of \(\mathrm{n}\)

• What this tells us: We're looking for a pattern where each height is half the previous height.

2. INFER the mathematical pattern

• This creates a geometric sequence where:
  • First term (after 1st bounce) = \(80 \times 0.5\)
  • Each subsequent term = previous term \(\times 0.5\)
• The pattern will be: \(\mathrm{h} = 80 \times (0.5)^\mathrm{n}\)

3. SIMPLIFY by calculating the first few bounces

• After 1st bounce: \(\mathrm{h_1} = 80 \times 0.5^1 = 40\) feet
• After 2nd bounce: \(\mathrm{h_2} = 80 \times (0.5)^2 = 80 \times 0.25 = 20\) feet
• After 3rd bounce: \(\mathrm{h_3} = 80 \times (0.5)^3 = 80 \times 0.125 = 10\) feet

4. APPLY CONSTRAINTS by testing answer choices

• Test each choice with \(\mathrm{n} = 1\) (should give \(\mathrm{h} = 40\)):
  • (A) \(\mathrm{h} = 40(0.5)^1 = 20\) ❌
  • (B) \(\mathrm{h} = 80(0.5)^0 = 80 \times 1 = 80\) ❌
  • (C) \(\mathrm{h} = 80(0.5)^1 = 40\) ✓
  • (D) \(\mathrm{h} = 80(2)^1 = 160\) ❌

• Only choice (C) works for our test case.

Answer: C

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students misinterpret 'after its nth bounce' and think \(\mathrm{n} = 1\) should give the initial height of 80 feet, not the height after the first bounce (40 feet).

This leads them to test answer choices incorrectly, expecting \(\mathrm{n} = 1\) to yield 80. When they see choice (B) gives \(\mathrm{h} = 80(0.5)^0 = 80\) for \(\mathrm{n} = 1\), they incorrectly think this is right.

This may lead them to select Choice B (\(80(0.5)^{\mathrm{n}-1}\)).

Second Most Common Error:

Poor INFER reasoning: Students recognize the pattern but confuse the starting point. They think the sequence should begin with 80 as the first term instead of recognizing that 80 is the drop height, not a bounce height.

This conceptual confusion about what the sequence represents causes them to get stuck and guess randomly among the choices.

The Bottom Line:

The key challenge is correctly interpreting what 'after its nth bounce' means - the first bounce creates \(\mathrm{h_1} = 40\), not \(\mathrm{h_1} = 80\). Students must carefully distinguish between the initial drop height and the heights achieved after bouncing.