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The table shows the balance of an investment account at the beginning of each year the account was held. Assuming that no other deposits have been made to the account, which statement describes the account's growth?

\begin{tabular}{|c|c|}
\hline Year & \begin{tabular}{c}
Account \\
Balance
\end{tabular} \\
\hline 1 & \$200.00 \\
\hline 2 & \$208.00 \\
\hline 3 & \$216.32 \\
\hline
\end{tabular}

A. The account is growing linearly at an annual interest rate of 4.00\%.

B. The account is growing exponentially at an annual interest rate of 4.00\%.

C. The account is growing linearly at an annual interest rate of 8.16\%.

D. The account is growing exponentially at an annual interest rate of 8.16\%.



Answer :

GinnyAnswer
Let's analyze the account growth step by step.

1. Balance at the beginning of each year:
- Year 1: $200.00
- Year 2: $208.00
- Year 3: $216.32

2. Calculate the annual growth rates:
- Growth rate from Year 1 to Year 2: \((\frac{208.00}{200.00}) - 1 = 0.04\)
- Growth rate from Year 2 to Year 3: \((\frac{216.32}{208.00}) - 1 = 0.04\)

3. Average the growth rates:
[tex]\[ \text{Average growth rate} = \frac{0.04 + 0.04}{2} = 0.04 \][/tex]

4. Convert to percentage:
[tex]\[ \text{Average growth rate percentage} = 0.04 \times 100 = 4.00\% \][/tex]

5. Check for exponential growth at 4.00% annual interest:
- Expected balance for Year 2: \(200.00 \times (1 + 0.04) = 208.00\)
- Expected balance for Year 3: \(208.00 \times (1 + 0.04) = 216.32\)

Since the actual balances match the expected balances:
- 208.00 matches 208.00 for Year 2
- 216.32 matches 216.32 for Year 3

This indicates the account is growing exponentially at an annual interest rate of 4.00%.

6. Conclusion:
- Given the calculations and balances matching perfectly for exponential growth at a 4.00% rate:

The correct statement is:
B. The account is growing exponentially at an annual interest rate of [tex]$4.00 \%$[/tex].

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