Questions: Find a bank account balance if the account starts with 1000, has an annual rate of 4%, and the money is left in the account for 25 years.

Transcript text: ponential Growth and Decay Activity Find a bank account balance if the account starts with $\$ 1000$, has an annual rate of $4 \%$, and the money is left in the account for 25 years. Answer. $\square$

Solution

Solution Steps

To find the bank account balance after 25 years with an initial amount of $1000 and an annual interest rate of 4%, we can use the formula for compound interest: \[ A = P \left(1 + \frac{r}{n}\right)^{nt} \] where:

$ A $ is the amount of money accumulated after n years, including interest.
$ P $ is the principal amount (the initial amount of money).
$ r $ is the annual interest rate (decimal).
$ n $ is the number of times that interest is compounded per year.
$ t $ is the time the money is invested for in years.

In this case, the interest is compounded annually, so $ n = 1 $.

Step 1: Identify the Variables

We start with the following values:

Principal amount $ P = 1000 $
Annual interest rate $ r = 0.04 $
Number of times interest is compounded per year $ n = 1 $
Time in years $ t = 25 $

Step 2: Apply the Compound Interest Formula

We use the compound interest formula: \[ A = P \left(1 + \frac{r}{n}\right)^{nt} \] Substituting the identified values: \[ A = 1000 \left(1 + \frac{0.04}{1}\right)^{1 \cdot 25} \]

Step 3: Calculate the Amount

Calculating the expression: \[ A = 1000 \left(1 + 0.04\right)^{25} = 1000 \left(1.04\right)^{25} \] Evaluating $ (1.04)^{25} $ gives approximately $ 2.6658 $. Thus: \[ A \approx 1000 \times 2.6658 = 2665.8363 \]