An initial amount of money was deposited into an investment account that earns interest compounded annually. The value of the...

1. TRANSLATE the problem information

• Given information:
  • Investment grows by 5% each year (compounded annually)
  • After 2 years, the value is $4,410
  • Need to find \(\mathrm{V(t)}\) = value after t years

2. INFER the mathematical approach

• This describes exponential growth, which follows the formula \(\mathrm{V(t) = P(1 + r)^t}\)
• We know the growth rate (\(\mathrm{r = 0.05}\)) and one point on the curve (\(\mathrm{t = 2}\), \(\mathrm{V = 4{,}410}\))
• Strategy: Use this information to find the initial principal P, then write the complete equation

3. Set up the exponential growth equation

• The growth factor is \(\mathrm{(1 + r) = 1 + 0.05 = 1.05}\)
• General form: \(\mathrm{V(t) = P(1.05)^t}\)
• We know that \(\mathrm{V(2) = 4{,}410}\)

4. SIMPLIFY to find the initial principal

• Substitute the known values: \(\mathrm{4{,}410 = P(1.05)^2}\)
• Calculate the exponent: \(\mathrm{(1.05)^2 = 1.1025}\) (use calculator)
• Solve for P: \(\mathrm{P = 4{,}410 \div 1.1025 = 4{,}000}\) (use calculator)

5. Write the complete equation

• With P = 4,000 and growth factor = 1.05
• The equation is: \(\mathrm{V(t) = 4000(1.05)^t}\)

Answer: B

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students misinterpret what the $4,410 represents, thinking it's the initial principal rather than the value after 2 years.

They assume P = 4,410 and create the equation \(\mathrm{V(t) = 4410(1.05)^t}\), leading them to select Choice D (\(\mathrm{V(t) = 4410(1.05)^t}\)).

Second Most Common Error:

Poor INFER reasoning: Students recognize they need exponential growth but confuse the rate with the growth factor, using 0.05 instead of 1.05 as the base.

This leads them to incorrectly write \(\mathrm{V(t) = 4410(0.05)^t}\) and select Choice C (\(\mathrm{V(t) = 4410(0.05)^t}\)).

The Bottom Line:

This problem requires careful attention to what each piece of given information represents and understanding the structure of exponential growth equations. The key insight is that the $4,410 is a result after growth has occurred, not the starting amount.