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Chapter 19: Problem 46

Find the indicated quantities for the appropriate arithmetic sequence. A bank loan of \(\$ 8000\) is repaid in annual payments of \(\$ 1000\) plus \(10 \%\) interest on the unpaid balance. What is the total amount of interest paid?

Short Answer

Expert verified
The total interest paid is $2865.

Step by step solution

01

Understanding the Problem

To solve this problem, recognize that each year involves paying a fixed amount and interest on the remaining loan balance. The problem requires calculating the total interest paid over the life of the loan.
02

Setting up the Sequence

The loan starts at $8000, and each year $1000 is paid off. After paying $1000, interest is applied to the remaining balance. The sequence of remaining loan balances will start at $8000 and decrease by $1000 each year until it is paid off.
03

Calculate Yearly Balances

Calculate the balance at the end of each year after the payment and interest: - Year 0: Start with $8000. - Year 1: Balance = $8000 - $1000 = $7000; interest = 0.10 x $7000 = $700; New Balance = $7000 + $700 = $7700. - Repeat calculations for subsequent years until the loan is paid off.
04

Calculate Annual Interest

- Year 1: Interest on $7000 = $700. - Year 2: Payoff from $7700 is $6700 (after a $1000 payment), interest on $6700 = $670; - Year 3: Payoff from $7370 is $6370, interest on $6370 = $637; - Continue this pattern until the balance is zero.
05

Summing Up Interests

Sum the yearly interests to find the total interest paid over the entire loan period. For each year, add the calculated interest to get the cumulative interest paid.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Loan Repayment
When you take out a bank loan, such as the $8000 mentioned here, you're expected to repay the amount over time, typically in regular installments. For this exercise, loan repayment is structured with annual payments of $1000. Each year, you reduce the principal balance of the loan by the amount you've repaid.

In simpler terms, to pay off an $8000 loan, you'd pay $1000 every year. After the first year, you'd only owe $7000. This pattern continues until the loan is completely paid off. This type of repayment creates what's called a sequence, particularly an arithmetic sequence, where each payment period's remaining balance forms part of a predictable pattern.
  • The initial loan amount: $8000
  • Annual payment: $1000
  • Loan decreases by $1000 annually
Interest Calculation
Interest calculation on a loan is crucial because it determines how much extra money you'll have paid by the time the loan is fully repaid. For this particular loan, the interest rate is 10% on the unpaid balance.

Let's explore how it works:
  • Year 1: You pay off $1000, leaving a $7000 balance. The interest charged will be 10% of $7000, which is $700.
  • Year 2: After another $1000 payment, the balance becomes $6700, and the interest for this year would be 10% of $6700, which is $670.
  • Continue the same pattern for the remaining years until the balance is zero.
The interest for each year is calculated based on the remaining balance after the annual payment has been made. Ensuring you understand this helps you realize how much more you need to pay beyond just the loan's principal amount.
Unpaid Balance
The unpaid balance is a key element in understanding loan repayment. It refers to the amount remaining on the loan after payments have been made each year. This balance is crucial for interest calculation because interest is charged on what's left unpaid.

For instance;
  • After your first $1000 payment, the unpaid balance drops to $7000 from the original $8000.
  • After paying the next $1000, it becomes $6700 and so on.
This process continues until the balance reaches zero. Knowing the unpaid balance at any point allows you to calculate the interest for that year accurately. The arithmetic sequence of the unpaid balance gives a clear picture of both your repayment progress and the associated interest each year. It's an essential part of managing and understanding the total cost of a loan.

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Most popular questions from this chapter

Find the indicated terms by use of the following information. The \(r+1\) term of the expansion of \((a+b)^{n}\) is given by $$\frac{n(n-1)(n-2) \cdots(n-r+1)}{r !} a^{n-r} b^{r}$$ The sixth term of \((\sqrt{a}-\sqrt{b})^{14}\)

Find the indicated quantities for the appropriate arithmetic sequence. In preparing a bid for constructing a new building, a contractor determines that the foundation and basement will cost \(\$ 605,000\) and the first floor will cost \(\$ 360,000\). Each floor above the first will cost \(\$ 15,000\) more than the one below it. How much will the building cost if it is to be 18 floors high?

Find any of the values of \(a_{1}, r, a_{n}, n,\) or \(S_{n}\) that are missing. $$r=-\frac{1}{2}, a_{n}=\frac{1}{8}, n=7$$

Find the indicated quantities for the appropriate arithmetic sequence. A person has a \(\$ 5000\) balance due on a credit card account that charges \(1 \%\) interest per month on the unpaid balance \(B_{n}\). Assuming no extra charges, if \(\$ 250\) is paid each month, find (a) a recursion formula [similar to Eq. (19.1)] for \(B_{n}\) and (b) \(B_{n}\) after two months.

Find the indicated quantities for the appropriate arithmetic sequence. The sequence of ships' bells is as follows: 12: 30 A.M. one bell is rung, and each half hour later one more bell is rung than the previous time until eight bells are rung. The sequence is then repeated starting at 4: 30 A.M., again until eight bells are rung. This pattern is followed throughout the day. How many bells are rung in one day?
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