\(\mathrm{A(n) = 500(1.08)^n}\). The function A defined above can be used to model the value in dollars of an investment...

1. TRANSLATE the problem information

• Given information:
  • Function: \(\mathrm{A(n) = 500(1.08)^n}\)
  • \(\mathrm{A(n)}\) represents investment value in dollars
  • \(\mathrm{n}\) represents number of 3-month quarters since initial investment

• What this tells us: We need to understand what happens when \(\mathrm{n}\) changes by 1 unit (which equals 3 months of time)

2. INFER the mathematical meaning

• In exponential functions \(\mathrm{A(n) = P(r)^n}\), the value \(\mathrm{r}\) is the growth factor
• Since our growth factor is \(\mathrm{1.08}\), let's break this down:
  • \(\mathrm{1.08 = 1 + 0.08}\)
  • This means each time \(\mathrm{n}\) increases by 1, the value gets multiplied by \(\mathrm{1.08}\)
  • Multiplying by \(\mathrm{1.08}\) is the same as increasing by \(\mathrm{8\%}\)

3. Connect the time period to the growth

• Since \(\mathrm{n}\) represents 3-month quarters:
  • When \(\mathrm{n}\) increases by 1, time increases by 3 months
  • When time increases by 3 months, the value increases by \(\mathrm{8\%}\)

4. Check against answer choices

• Looking for: "For every increase in time by 3 months, the investment value increases by 8%"
• This matches choice (D) exactly

Answer: D

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students misinterpret what \(\mathrm{n}\) represents, thinking it means individual months rather than 3-month quarters.

If they think \(\mathrm{n}\) represents months, they might reason: "When \(\mathrm{n}\) increases by 1 (thinking this means 1 month), the value increases by 8%." This leads them to select Choice B (For every increase in time by 1 month, the investment value increases by 8%).

Second Most Common Error:

Conceptual confusion about exponential vs. linear growth: Students might focus on the coefficient 500 and think the growth is linear.

They might incorrectly think: "The value starts at 500, so it must increase by $500 each time period." This could lead them to select Choice C (For every increase in time by 3 months, the investment value increases by $500).

The Bottom Line:

This problem tests whether students can correctly interpret the components of an exponential model and connect the mathematical structure to real-world time periods. The key insight is understanding that the growth factor applies to the time unit specified by the variable definition.