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




Year		Account Balance

1		\$1,200.00

2		\$1,260.00

3		\$1,323.00

4		\$1,389.15





AThe account is growing exponentially at an annual interest rate of 10.25\%.
BThe account is growing linearly at an annual interest rate of 5.00\%.
CThe account is growing exponentially at an annual interest rate of 5.00\%.
DThe account is growing linearly at an annual interest rate of 10.25\%.
EThe account is growing exponentially at an annual interest rate of 15.76\%.

Verified Answer :

  • The account is not growing linearly, as the year-over-year differences (\(\$60.00, \$63.00, \$66.15\)) are not constant.
  • The account is growing exponentially, as the year-over-year ratios (\(1.05, 1.05, 1.05\)) are constant.
  • The annual growth factor \(r = 1.05\) corresponds to an interest rate of \(i = r - 1 = 0.05\), which is \(5.00\%\).
  • The final answer is \(\boxed{\text{The account is growing exponentially at an annual interest rate of } 5.00\%}\).

Explanation

  1. Analyzing Growth Patterns

To determine how the account balance is growing, we analyze the data provided for four consecutive years. Let \(A_n\) be the account balance at the start of year \(n\). We have:

  • Year 1: \(A_1 = \$1,200.00\)
  • Year 2: \(A_2 = \$1,260.00\)
  • Year 3: \(A_3 = \$1,323.00\)
  • Year 4: \(A_4 = \$1,389.15\)

First, we check for linear growth. A linear growth model implies a constant difference \(d\) between consecutive years, such that \(A_{n+1} - A_n = d\).

  • \(A_2 - A_1 = \$1,260.00 - \$1,200.00 = \$60.00\)
  • \(A_3 - A_2 = \$1,323.00 - \$1,260.00 = \$63.00\)
  • \(A_4 - A_3 = \$1,389.15 - \$1,323.00 = \$66.15\)

Since the differences are not constant, the growth is not linear.

  1. Verifying Exponential Growth and Calculating the Interest Rate

Next, we check for exponential growth. An exponential growth model implies a constant ratio \(r\) between consecutive years, such that \(\frac{A_{n+1}}{A_n} = r\).

  • \(\frac{A_2}{A_1} = \frac{\$1,260.00}{\$1,200.00} = 1.05\)
  • \(\frac{A_3}{A_2} = \frac{\$1,323.00}{\$1,260.00} = 1.05\)
  • \(\frac{A_4}{A_3} = \frac{\$1,389.15}{\$1,323.00} = 1.05\)

Since the ratio is constant at \(1.05\), the account follows an exponential growth model. The annual growth factor \(r\) is \(1.05\), which is related to the annual interest rate \(i\) by the formula \(r = 1 + i\). Thus, \(1 + i = 1.05\), which gives \(i = 0.05\), or \(5.00\%\).

  1. Conclusion

We have determined that the account is growing exponentially at an annual interest rate of \(5.00\%\). Comparing this to the given options, the correct statement is: The account is growing exponentially at an annual interest rate of \(5.00\%\). The final result is \(\boxed{\text{The account is growing exponentially at an annual interest rate of } 5.00\%}\).

Examples

Understanding whether an investment grows linearly or exponentially is crucial for long-term financial planning. For instance, if you deposit money into a high-yield savings account, it usually grows exponentially because you earn interest on your interest (compounding). On the other hand, if you were to receive a fixed amount of cash from a family member every year, your total savings would grow linearly. Recognizing these patterns helps you predict your future wealth more accurately when deciding between different savings options.

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