Question

Write a program that prints the balance of an account after the first, second, and third year. The account has an initial balance of \(\$ 1,000\) and carns 5 percent interest per year.

Short answer

After 1 year: $1,050; after 2 years: $1,102.50; after 3 years: $1,157.63.

Step-by-step solution

Define Initial Balance and Interest Rate

Start by defining variables for the initial balance and the interest rate. In this case, the initial balance is $1,000 and the annual interest rate is 5%.

Calculate Balance after First Year

To determine the balance after the first year, apply the interest rate to the initial balance. This can be calculated as: \[ \text{Balance after 1st Year} = \text{Initial Balance} \times (1 + \text{Interest Rate}) \] Since the interest rate is 5%, the calculation is: \[ 1000 \times 1.05 = 1050 \] So, the balance after the first year is $1,050.

Calculate Balance after Second Year

Use the balance after the first year to calculate the balance after the second year with the same interest rate. The formula is: \[ \text{Balance after 2nd Year} = \text{Balance after 1st Year} \times (1 + \text{Interest Rate}) \] Thus: \[ 1050 \times 1.05 = 1102.5 \] The balance after the second year is $1,102.50.

Calculate Balance after Third Year

Now calculate the balance after the third year using the balance from the second year. Again, apply the interest rate: \[ \text{Balance after 3rd Year} = \text{Balance after 2nd Year} \times (1 + \text{Interest Rate}) \] Thus: \[ 1102.5 \times 1.05 = 1157.625 \] The balance after the third year is $1,157.63 (rounded to two decimal places).

Write and Execute the Program

Write a simple program in Python that uses these calculations to print the balance after each year. Here's how you can implement it: ```python initial_balance = 1000 interest_rate = 0.05 # Calculate and print balance after each year year_1_balance = initial_balance * (1 + interest_rate) year_2_balance = year_1_balance * (1 + interest_rate) year_3_balance = year_2_balance * (1 + interest_rate) print('Balance after 1st year: $', round(year_1_balance, 2)) print('Balance after 2nd year: $', round(year_2_balance, 2)) print('Balance after 3rd year: $', round(year_3_balance, 2)) ``` This program will output the balance for each year.

Key concepts

Python Programming

Python is a powerful and versatile programming language that's incredibly beginner-friendly. If you've never programmed before, Python is a great place to start. For this particular exercise, you'll be writing a simple program to calculate future account balances using compound interest. This involves creating variables for both your initial balance and interest rate. Variables in Python act as containers that store data values and are defined using a straightforward syntax.

In this example, you set the `initial_balance` to 1000 and the `interest_rate` to 0.05 (which represents the percent increase as a decimal). The equation for calculating the balance after interest is applied is:

• Balance after each year can be calculated using the formula \( \text{Balance} = \text{Initial Balance} \times (1 + \text{Interest Rate}) \)
• The `print` function in Python allows you to display your results. It's used here to output each year's balance clearly formatted for others to read.
• The `round` function helps limit the results to two decimal places, which is standard for financial transactions.

You'll use these simple arithmetic and functioning techniques to implement your calculations and see the power Python holds in automating routine calculations.

Financial Literacy

Understanding compound interest is a vital aspect of financial literacy. It allows you to grasp how your savings or investments grow over time. In this exercise, the principle of compound interest is applied, where interest earned on an account is added to the initial balance each year, resulting in an ever-increasing amount of interest earned. This exponential growth is key to wealth-building.

Here's how it works:

• **Initial Principal**: This is your starting amount, which in this case is $1,000.
• **Interest Rate**: The percentage at which your balance increases annually - 5% in this example. You convert this percent to a decimal (0.05) for easy calculation.
• **Compounding**: By reinvesting earnings (interest added each year), the account grows faster over multiple years.
• **Balance Growth**: Each year's balance builds on the previous year's balance, not merely the original principal, leading to more growth over time.

This exercise shows the snowballing effect of compounded earnings, emphasizing why starting to save and invest early can lead to significant financial benefits.

Basic Arithmetic Operations

In the realm of computing future account balances, basic arithmetic operations are essential. These operations include addition, multiplication, and the use of parentheses, which are foundational in understanding mathematical expressions in programming.

Here's a breakdown of their use in this scenario:

• **Multiplication** is used to apply the interest rate to your current balance year-over-year, increasing your total account balance.
• **Addition** is implicit in the multiplication process when you implement \( 1 + \text{Interest Rate} \). This factor ensures the interest is added each year.
• **Use of Parentheses**: Parentheses let you control the order of operations, ensuring that the addition and multiplication occur in the logical sequence necessary for calculating compound interest.

Practicing with these operations in Python via this exercise reinforces your arithmetic skills and illustrates their practical applications in finance. These operations underpin more complex algorithms and financial models you'll encounter later, serving as a building block for more advanced computational tasks.