Problem 5 - Algebra 2

Question

BANKING For Exercises 5 and \(6,\) use the following information. Rita has deposited 51000 in a bank account. At the end of each year, the bank posts 3\(\%\) interest to her account, but then takes out a \(\$ 10\) annual fee. Let \(b\) be the amount Rita deposited. Write a recursive equation for the balance \(b_{n}\) in her account at the end of \(n\) years.

Step-by-Step Solution

Answer

Recursive equation: \( b_n = 1.03 \times b_{n-1} - 10 \). Initial condition: \( b_0 = 51000 \).

Step 1: Understand the Components

Rita's account balance undergoes two main changes each year: interest is added, and a fee is deducted. The interest is a percentage of the current balance, while the fee is a fixed amount.

Step 2: Calculate the Interest

The interest added to the account each year is 3%. This means each year, the balance increases by 3% of the current balance. If the balance at the end of year \(n-1\) is \(b_{n-1}\), then the interest added is \(0.03 \times b_{n-1}\).

Step 3: Deduct the Fee

Each year, after adding interest, the bank deducts a \(\$10\) fee from the account.

Step 4: Formulate the Recursive Equation

To form a recursive equation, express the balance at the end of year \(n\), \(b_n\), based on the balance from the previous year, \(b_{n-1}\). We have:\[ b_n = b_{n-1} + 0.03 \times b_{n-1} - 10 \]This equation takes the previous balance, adds the interest, and then subtracts the fee.

Step 5: Simplify the Equation

Combine terms to simplify:\[ b_n = 1.03 \times b_{n-1} - 10 \]This is the recursive relationship for Rita's account.

Key Concepts

Banking Interest

Banking interest is the process by which banks add a percentage of the current balance to an account. This percentage is called the interest rate. For Rita, the bank posts 3% interest annually. Let's break down what this means.

• If Rita's account balance, let's say, at the beginning of the year is \( b_{n-1} \), then by the end of the year, the bank calculates 3% of that balance as interest.
• The formula for calculating interest is: \( 0.03 \times b_{n-1} \).
• This interest is added directly to the account, increasing the total balance.

Understanding banking interest is crucial as it directly impacts how much your money grows over time. The higher the interest rate, the more your balance can increase annually.

Annual Fees

Annual fees are charges that banks apply to accounts once every year. For Rita, this fee amounts to \\(10. Let's see how these fees work and how they influence the account.

• After calculating and adding the interest to Rita's account, the bank then deducts \\)10 as an annual fee.
• The fee is a constant, meaning it's always \$10 regardless of the account balance.

These fees reduce the growth of an account's balance because they are taken out after interest is added. Always keep an eye on such charges as they can significantly impact long-term savings, eroding some of the benefits gained from interest.

Balance Calculation

Calculating the balance in an account involves adding interest first, and then subtracting any fees. For Rita, this process happens yearly. Below is how her balance is calculated.

• Start with the balance from the previous year, \( b_{n-1} \).
• Add 3% interest: \( 0.03 \times b_{n-1} \).
• Subtract the \$10 fee.

So the formula becomes: \[ b_n = 1.03 \times b_{n-1} - 10 \].

This equation incorporates both the increase due to interest and the decrease due to fees, giving you the new balance at the end of year \( n \). Calculating your balance with this method helps track your savings accurately over time.

Algebraic Recurrence

Algebraic recurrence describes a sequence where each term is defined in terms of one or more of its predecessors. In Rita's case, we're examining how her bank balance evolves over time using a recursive formula.

• The recursive formula provided is: \[ b_n = 1.03 \times b_{n-1} - 10 \].
• Here, \( b_n \) (the balance at year \( n \)) is derived from \( b_{n-1} \) (the balance from the previous year).
• This formula captures both the compounding interest and the deduction of annual fees.

By repeatedly applying the recursive formula, you can predict future balances, plan for savings targets, or identify how long a balance will last under given conditions. This type of analytical approach is powerful in not just banking, but in any field involving compound growth or decay.