unit 3 • finance
topic c: loans and financed purchases
practice 3.6: recursion and sequences—payment plans
use what you know about sequences to complete each problem.
1. what is the fourth term in the sequence given by the formula $a_n = 10n - 12$?
2. what is the fourth term in the sequence given by the formula $a_n = a_{n - 1} + 3$ if $a_1 = -4$?
3. what is the third term in the sequence given by the formula $a_n = a_{n - 1} \bullet 4$ if $a_1 = 2$?
4. what is the fourth term in the sequence given by the formula $a_n = 13(2)^{n - 1}$?
use the given information to complete problems 5–7.
muna is purchasing a car. she is considering a payment plan that charges 15% apr, with monthly payments of $350. the list price of the car is $18,500.
5. what is the monthly interest rate?
6. find a recursive formula to model the new balance in terms of the previous balance.
7. generate the first 5 rows of an amortization table based on this scenario.

Question

Accepted Answer

### Problem 1
# Explanation:
## Step 1: Identify $ n $ for the fourth term
For the fourth term, $ n = 4 $.
## Step 2: Substitute $ n = 4 $ into the formula $ a_n = 10n - 12 $
Substitute $ n = 4 $ into $ a_n = 10n - 12 $:
$ a_4 = 10(4) - 12 $
$ a_4 = 40 - 12 $
$ a_4 = 28 $

# Answer:
The fourth term is $ 28 $.

### Problem 2
# Explanation:
## Step 1: Find $ a_2 $ using $ a_1 = -4 $ and $ a_n = a_{n - 1} + 3 $
Substitute $ n = 2 $, $ a_1 = -4 $:
$ a_2 = a_{1} + 3 = -4 + 3 = -1 $
## Step 2: Find $ a_3 $ using $ a_2 = -1 $
Substitute $ n = 3 $, $ a_2 = -1 $:
$ a_3 = a_{2} + 3 = -1 + 3 = 2 $
## Step 3: Find $ a_4 $ using $ a_3 = 2 $
Substitute $ n = 4 $, $ a_3 = 2 $:
$ a_4 = a_{3} + 3 = 2 + 3 = 5 $

# Answer:
The fourth term is $ 5 $.

### Problem 3
# Explanation:
## Step 1: Find $ a_2 $ using $ a_1 = 2 $ and $ a_n = a_{n - 1} \cdot 4 $
Substitute $ n = 2 $, $ a_1 = 2 $:
$ a_2 = a_{1} \cdot 4 = 2 \cdot 4 = 8 $
## Step 2: Find $ a_3 $ using $ a_2 = 8 $
Substitute $ n = 3 $, $ a_2 = 8 $:
$ a_3 = a_{2} \cdot 4 = 8 \cdot 4 = 32 $
## Step 3: Find the third term (wait, the question asks for the third term? Wait, the problem says "third term" but the formula is $ a_n = a_{n - 1} \cdot 4 $, $ a_1 = 2 $. Wait, let's recheck:

Wait, the problem is: "What is the third term in the sequence given by the formula $ a_n = a_{n - 1} \cdot 4 $ if $ a_1 = 2 $?"

So:
- $ a_1 = 2 $
- $ a_2 = a_1 \cdot 4 = 2 \cdot 4 = 8 $
- $ a_3 = a_2 \cdot 4 = 8 \cdot 4 = 32 $

# Answer:
The third term is $ 32 $.

### Problem 4
# Explanation:
## Step 1: Identify $ n $ for the fourth term
For the fourth term, $ n = 4 $.
## Step 2: Substitute $ n = 4 $ into $ a_n = 13(2)^{n - 1} $
Substitute $ n = 4 $:
$ a_4 = 13(2)^{4 - 1} $
$ a_4 = 13(2)^3 $
$ a_4 = 13 \cdot 8 $
$ a_4 = 104 $

# Answer:
The fourth term is $ 104 $.

### Problem 5
# Explanation:
## Step 1: Recall APR to monthly rate conversion
APR (Annual Percentage Rate) is annual, so to find the monthly rate, divide by 12 (number of months in a year).
Monthly rate $ = \frac{\text{APR}}{12} $
## Step 2: Substitute APR = 15%
Monthly rate $ = \frac{15\%}{12} = 1.25\% $ (or as a decimal, $ 0.0125 $, or as a multiplier $ 1 + 0.0125 = 1.0125 $)

# Answer:
The monthly interest rate is $ 1.25\% $ (or $ 0.0125 $ as a decimal, or $ 1.0125 $ as a multiplier).

### Problem 6
# Explanation:
## Step 1: Define the recursive formula structure
Let $ a_n $ be the new balance (after $ n $ months), and $ a_{n - 1} $ be the previous balance.
- First, apply the monthly interest: $ a_{n - 1} \times (1 + \text{monthly rate}) $
- Then, subtract the monthly payment: $ \$350 $

## Step 2: Substitute the monthly rate
Monthly rate is $ 1.25\% = 0.0125 $, so the multiplier is $ 1 + 0.0125 = 1.0125 $.

Thus, the recursive formula is:
$ a_n = 1.0125 \cdot a_{n - 1} - 350 $, with $ a_0 = 18500 $ (initial balance, the car price).

# Answer:
The recursive formula is $ a_n = 1.0125a_{n - 1} - 350 $ (with $ a_0 = 18500 $).

### Problem 7
# Explanation:
## Step 1: Define the amortization table columns
Columns: Month ($ n $), Previous Balance ($ a_{n - 1} $), Interest ($ 0.0125 \cdot a_{n - 1} $), Payment ($ \$350 $), New Balance ($ a_n = 1.0125a_{n - 1} - 350 $)

## Step 2: Calculate each row (Month 1 to Month 5)
- **Month 1**:
  $ a_0 = 18500 $
  Interest: $ 0.0125 \times 18500 = 231.25 $
  New Balance: $ 1.0125 \times 18500 - 350 = 18500 + 231.25 - 350 = 18381.25 $

- **Month 2**:
  $ a_1 = 18381.25 $
  Interest: $ 0.0125 \times 18381.25 \approx 229.77 $
  New Balance: $ 1.0125 \times 18381.25 - 350 \approx 18381.25 + 229.77 - 350 = 18261.02 $

- **Month 3**:
  $ a_2 = 18261.02 $
  Interest: $ 0.0125 \times 18261.02 \approx 228.26 $
  New Balance: $ 1.0125 \times 18261.02 - 350 \approx 18261.02 + 228.26 - 350 = 18139.28 $

- **Month 4**:
  $ a_3 = 18139.28 $
  Interest: $ 0.0125 \times 18139.28 \approx 226.74 $
  New Balance: $ 1.0125 \times 18139.28 - 350 \approx 18139.28 + 226.74 - 350 = 18016.02 $

- **Month 5**:
  $ a_4 = 18016.02 $
  Interest: $ 0.0125 \times 18016.02 \approx 225.20 $
  New Balance: $ 1.0125 \times 18016.02 - 350 \approx 18016.02 + 225.20 - 350 = 17891.22 $

## Amortization Table (first 5 rows):

| Month ($ n $) | Previous Balance (\$) | Interest (\$) | Payment (\$) | New Balance (\$) |
|-----------------|-----------------------|---------------|--------------|------------------|
| 1               | 18500.00              | 231.25        | 350.00       | 18381.25         |
| 2               | 18381.25              | 229.77        | 350.00       | 18261.02         |
| 3               | 18261.02              | 228.26        | 350.00       | 18139.28         |
| 4               | 18139.28              | 226.74        | 350.00       | 18016.02         |
| 5               | 18016.02              | 225.20        | 350.00       | 17891.22         |

# Answer (for the table):
| Month ($ n $) | Previous Balance (\$) | Interest (\$) | Payment (\$) | New Balance (\$) |
|-----------------|-----------------------|---------------|--------------|------------------|
| 1               | 18500.00              | 231.25        | 350.00       | 18381.25         |
| 2               | 18381.25              | 229.77        | 350.00       | 18261.02         |
| 3               | 18261.02              | 228.26        | 350.00       | 18139.28         |
| 4               | 18139.28              | 226.74        | 350.00       | 18016.02         |
| 5               | 18016.02              | 225.20        | 350.00       | 17891.22         |

Sovi.AI - AI Math Tutor

QUESTION IMAGE

Question

Explanation:

Problem 1

Step 1: Identify \( n \) for the fourth term

Step 2: Substitute \( n = 4 \) into the formula \( a_n = 10n - 12 \)

Step 1: Find \( a_2 \) using \( a_1 = -4 \) and \( a_n = a_{n - 1} + 3 \)

Step 2: Find \( a_3 \) using \( a_2 = -1 \)

Step 3: Find \( a_4 \) using \( a_3 = 2 \)

Step 1: Find \( a_2 \) using \( a_1 = 2 \) and \( a_n = a_{n - 1} \cdot 4 \)

Step 2: Find \( a_3 \) using \( a_2 = 8 \)

Step 3: Find the third term (wait, the question asks for the third term? Wait, the problem says "third term" but the formula is \( a_n = a_{n - 1} \cdot 4 \), \( a_1 = 2 \). Wait, let's recheck:

Answer:

Problem 2