Questions: Kevin examines the following data, which shows the balance in an investment account. - Year 1: 2,000.00 - Year 2: 2,120.00 - Year 3: 2,247.20 - Year 5: 2,382.03 What is the formula for the geometric sequence represented by the data above? - (an=2000(1.6)^n-1) - (an=2000(1.06)^n-1) - (an=2000(1.6)^n)

Solution Steps

To find the formula for the geometric sequence, we need to determine the common ratio between consecutive terms. We can do this by dividing the balance of one year by the balance of the previous year. Once we have the common ratio, we can use the formula for a geometric sequence, \( a_n = a_1 \cdot r^{n-1} \), where \( a_1 \) is the initial term and \( r \) is the common ratio.

To find the common ratio \( r \) of the geometric sequence, we divide the balance of year 2 by the balance of year 1: \[ r = \frac{2120.00}{2000.00} = 1.06 \]

The initial term \( a_1 \) of the sequence is the balance in year 1: \[ a_1 = 2000.00 \]

Using the common ratio and the initial term, the formula for the geometric sequence can be expressed as: \[ a_n = 2000.00 \cdot (1.06)^{n-1} \]

We compare the derived formula with the provided options:

1. \( a_n = 2000(1.6)^{n-1} \) - False
2. \( a_n = 2000(1.06)^{n-1} \) - True
3. \( a_n = 2000(1.6)^{n} \) - False

Final Answer