In December Corey took out a 1000-dollar loan with monthly interest rate 0.7 percent. In order to pay back the loan, Corey has been paying $87.17 a month since January. Explain why the sequence defined recursively x0=1000 and xn=1.007(xn-1) for positive n describes the balance of Corey's debt throughout the repayment of the loan.
Well, isn't Corey's debt a real charmer? This recursive sequence is like a little dance between Corey and his debt. Let me break it down for you.
First, we start with x0 = 1000, which represents Corey's initial debt of $1000. Now, imagine each term in the sequence as a month passing by. Pretty dramatic, huh?
In each subsequent month, Corey's debt is determined by multiplying the previous month's balance (xn-1) by 1.007, which accounts for the monthly interest rate of 0.7 percent. It's like Corey's debt is throwing a little "interest" party every month.
So, each term in the sequence represents the balance of Corey's debt for a specific month. Let's say we want to find out Corey's debt after 2 months. We plug in n=2 into the equation:
x2 = 1.007(x1)
To find x1, we substitute n=1 into the equation:
x1 = 1.007(x0)
And since x0 = 1000, we have:
x1 = 1.007(1000)
After calculating, we find:
x1 = 1007
Now, let's go back to finding x2:
x2 = 1.007(x1)
= 1.007(1007)
After crunching the numbers, we get:
x2 ≈ 1014.049
Voila! x2 represents the balance of Corey's debt after 2 months. See how this recursive sequence keeps track of Corey's debt month by month? It's like following Corey's financial journey with a dash of mathematical pizzazz.
The sequence defined recursively, xn = 1.007(xn-1), describes the balance of Corey's debt throughout the repayment of the loan because it takes into account the monthly interest rate of 0.7 percent.
Let's break down the explanation step-by-step:
1. Initial step: Corey took out a $1000 loan in December, which means that his debt can be represented as x0 = 1000.
2. Monthly interest rate: Corey has to pay interest on the remaining balance of the loan each month. The monthly interest rate is 0.7 percent, which can be written as 0.007 in decimal form.
3. Monthly payment: Corey has been paying $87.17 each month to repay the loan.
4. Recursive formula: The recursive formula xn = 1.007(xn-1) represents the balance of Corey's debt after each month's payment.
5. Explanation of the formula: The formula calculates the balance of Corey's debt for the nth month by multiplying the previous month's balance (xn-1) by 1.007. This takes into account the monthly interest rate of 0.7 percent, which increases the balance.
By following this recursive formula, the value of x1 is obtained by multiplying 1000 (x0) by 1.007. This represents Corey's debt after one month of payment.
Similarly, the value of x2 is obtained by multiplying x1 by 1.007, representing Corey's debt after two months of payment. This process continues for every positive integer n, calculating the balance of Corey's debt after each month's payment.
In summary, the recursive formula xn = 1.007(xn-1) allows us to calculate Corey's debt balance throughout the repayment of the loan, taking into account the monthly interest rate of 0.7 percent.
The sequence defined recursively as x0=1000 and xn=1.007(xn-1) for positive n describes the balance of Corey's debt throughout the repayment of the loan because it represents the concept of compound interest.
To understand this, let's break down the recursive formula.
- x0 represents the initial balance, which is $1000. This is the amount of the loan that Corey took out in December.
- xn represents the balance at month n. In other words, it tells us how much money Corey owes at the end of month n.
- xn-1 represents the balance at the previous month (n-1).
The recursive formula xn=1.007(xn-1) can be interpreted as follows:
- Take the balance at the previous month (xn-1).
- Multiply it by the monthly interest rate of 0.7 percent, which is equivalent to 1.007.
- The result represents the balance at the current month (xn).
By repeating this process every month, Corey's debt balance grows due to the effect of compound interest. The interest from each previous month is added to the balance, and then that total becomes the starting point for the next calculation.
Thus, the recursive sequence xn=1.007(xn-1) accurately describes the balance of Corey's debt throughout the repayment of the loan, taking into account the monthly interest rate and compounding effects.