Questions: 1,900 is deposited quarterly for 10 years at 4% per year FV=

Transcript text: answer to the nearest cent.) $\$ 1,900$ is deposited quarterly for 10 years at $4 \%$ per year \[ F V=\$ \] $\square$

Solution

Solution Steps

To solve this problem, we need to calculate the future value (FV) of an annuity with regular deposits. The formula for the future value of an annuity compounded quarterly is:

\[ FV = P \times \left( \frac{(1 + r/n)^{nt} - 1}{r/n} \right) \]

where:

$ P $ is the deposit amount per period ($1,900)
$ r $ is the annual interest rate (4% or 0.04)
$ n $ is the number of times the interest is compounded per year (quarterly, so 4)
$ t $ is the number of years (10)

Step 1: Identify Variables

We are given the following variables for the future value of an annuity:

$ P = 1900 $ (the deposit amount per period)
$ r = 0.04 $ (the annual interest rate)
$ n = 4 $ (the number of times interest is compounded per year)
$ t = 10 $ (the number of years)

Step 2: Apply the Future Value Formula

The future value $ FV $ of the annuity can be calculated using the formula: \[ FV = P \times \left( \frac{(1 + \frac{r}{n})^{nt} - 1}{\frac{r}{n}} \right) \]

Step 3: Substitute Values and Calculate

Substituting the values into the formula: \[ FV = 1900 \times \left( \frac{(1 + \frac{0.04}{4})^{4 \times 10} - 1}{\frac{0.04}{4}} \right) \] Calculating this gives: \[ FV = 1900 \times \left( \frac{(1 + 0.01)^{40} - 1}{0.01} \right) \] \[ FV = 1900 \times \left( \frac{(1.01)^{40} - 1}{0.01} \right) \] After performing the calculations, we find: \[ FV \approx 92884.11 \]