Problem 51 - Algebra 2

Question

FINANCE Regina pays \(\$ 50\) each month on a credit card that charges 1.6\(\%\) interest monthly. She has a balance of \(\$ 700 .\) The balance at the beginning of the \(n\) th month is given by \(f(n)=f(n-1)+0.016 f(n-1)-50 .\) Find the balance at the beginning of the first five months. No additional charges are made on the card. (Hint: \(f(1)=700 )\)

Step-by-Step Solution

Answer

The balances at the start of the first five months are 700, 661.2, 621.7792, 581.7276672, and 541.0353098752 dollars.

Step 1: Understand the Given Formula

The formula given is \(f(n) = f(n-1) + 0.016 \times f(n-1) - 50\). This describes Regina's balance on her credit card at the start of the \(n\)th month. It takes last month's balance \(f(n-1)\), adds the interest (1.6\% of last month's balance), and then subtracts the payment of \$50.

Step 2: Calculate the Balance at the Start of the 1st Month

According to the hint, \(f(1) = 700\), which means Regina's initial balance at the start of the first month is \$700.

Step 3: Calculate the Balance at the Start of the 2nd Month

Use the formula: \[f(2) = f(1) + 0.016 \times f(1) - 50\]Substitute \(f(1) = 700\):\[f(2) = 700 + 0.016 \times 700 - 50\]\[f(2) = 700 + 11.2 - 50 = 661.2\]

Step 4: Calculate the Balance at the Start of the 3rd Month

Use the formula: \[f(3) = f(2) + 0.016 \times f(2) - 50\]Substitute \(f(2) = 661.2\):\[f(3) = 661.2 + 0.016 \times 661.2 - 50\]\[f(3) = 661.2 + 10.5792 - 50 = 621.7792\]

Step 5: Calculate the Balance at the Start of the 4th Month

Use the formula: \[f(4) = f(3) + 0.016 \times f(3) - 50\]Substitute \(f(3) = 621.7792\):\[f(4) = 621.7792 + 0.016 \times 621.7792 - 50\]\[f(4) = 621.7792 + 9.9484672 - 50 = 581.7276672\]

Step 6: Calculate the Balance at the Start of the 5th Month

Use the formula: \[f(5) = f(4) + 0.016 \times f(4) - 50\]Substitute \(f(4) = 581.7276672\):\[f(5) = 581.7276672 + 0.016 \times 581.7276672 - 50\]\[f(5) = 581.7276672 + 9.3076426752 - 50 = 541.0353098752\]

Key Concepts

Financial Literacy

Understanding financial literacy concepts is crucial not only in theoretical exercises but also in real-life decision making. It equips individuals with the knowledge to make informed financial choices and manage their resources effectively. One of the core principles of financial literacy is an understanding of how debt works. This includes knowing how interest accumulates over time and how regular payments can affect the overall amount owed.

In this particular exercise, Regina's scenario shows how a consistent monthly payment can gradually reduce debt, even as interest continually grows. By understanding how interest charges are applied monthly and how making consistent payments helps decrease the debt, individuals can better plan their financial strategies. Being financially literate helps in making smart decisions when using credit cards, planning for big expenses, or saving for future goals.

• Recognize the importance of consistent payments.
• Understand how interest impacts your debt.
• Plan your finance effectively by understanding your liabilities and payments.

This knowledge not only applies to credit cards but also to other types of loans and financial products, enhancing one’s capability to manage finances efficiently.

Recurrence Relations

Recurrence relations are useful mathematical tools for solving problems where something is defined in terms of its previous terms. In the context of financial tasks like Regina's credit card balance, a recurrence relation helps quickly calculate the evolving state of her balance month by month.

The formula given, \[f(n) = f(n-1) + 0.016 f(n-1) - 50\]explains how each month's balance is derived from the previous month's figures, incorporating both the interest applied and the payment made. This type of relation provides a systematic approach to compute values without recalculating entire scenarios from scratch each time. Understanding this can greatly simplify calculations in financial contexts.

Key aspects of working with recurrence relations include:

• Identifying the base case, which in Regina's case starts with her initial balance.
• Breaking down complex calculations into simpler, smaller steps manageable by predefined formulas.
• Gaining insights into how these small, iterative changes reflect in the long term.

This approach is incredibly powerful, providing a robust framework for tackling numerous similar financial calculations.

Interest Calculation

Interest calculation is a pivotal concept in finance, influencing how quickly debts can grow and how efficiently they can be paid off. In Regina's case, the monthly interest of 1.6% significantly impacts her outstanding balance, despite her efforts to pay it down each month.

The way interest is calculated monthly means that it compounds over time, adding to the principal if not fully paid off. Regina's scenario demonstrates this process: \[f(n) = f(n-1) + 0.016 f(n-1)\]illustrates how her balance grows each month before her \$50 payment is deducted.

A few key takeaways about interest calculation include:

• Understanding the rate and how often it is applied is essential for accurate financial planning.
• Compounding interest impacts debt reduction and timeframes.
• Strategies to minimize interest include increasing payment amounts or reducing the outstanding balance faster.

Mastering interest calculation helps in creating more efficient payment strategies, ultimately aiding in faster debt reduction and better overall financial health.