Question
On Roy’s first birthday, his grandfather placed $100 in a bank account. On Roy’s second birthday, his grandfather placed $80 in the account and continued to place $80 each passing year. The arithmetic sequence 100, 180, 260, 340,… indicates the account balance for the first four years. Create a function that models this situation.(1 point)
Responses
f(n)=80(100)n−1
f left parenthesis n right parenthesis equals 80 left parenthesis 100 right parenthesis superscript n minus 1 baseline
f(n)=80+(n−1)100
f left parenthesis n right parenthesis equals 80 plus left parenthesis n minus 1 right parenthesis 100
f(n)=100+80(n)
f left parenthesis n right parenthesis equals 100 plus 80 left parenthesis n right parenthesis
f(n)=100+(n−1)80
Responses
f(n)=80(100)n−1
f left parenthesis n right parenthesis equals 80 left parenthesis 100 right parenthesis superscript n minus 1 baseline
f(n)=80+(n−1)100
f left parenthesis n right parenthesis equals 80 plus left parenthesis n minus 1 right parenthesis 100
f(n)=100+80(n)
f left parenthesis n right parenthesis equals 100 plus 80 left parenthesis n right parenthesis
f(n)=100+(n−1)80
Answers
GPT-4o mini
To model the account balance for Roy over the years, we need to consider the initial deposit and the subsequent annual deposits. The first deposit of $100 occurs on his first birthday, and every year after that until his nth birthday, $80 is deposited.
1. At \( n = 1 \) (Roy's first birthday), the balance is $100.
2. At \( n = 2 \) (Roy's second birthday), the balance is \( 100 + 80 = 180 \).
3. At \( n = 3 \) (Roy's third birthday), the balance is \( 100 + 80 + 80 = 260 \).
4. At \( n = 4 \) (Roy's fourth birthday), the balance is \( 100 + 80 + 80 + 80 = 340 \).
From the pattern, we observe that with each passing year, Roy's account balance increases by $80. The balance can be expressed as follows:
- For \( n = 1 \): \( 100 + 0 \cdot 80 = 100 \)
- For \( n = 2 \): \( 100 + 1 \cdot 80 = 180 \)
- For \( n = 3 \): \( 100 + 2 \cdot 80 = 260 \)
- For \( n = 4 \): \( 100 + 3 \cdot 80 = 340 \)
The general formula for the balance on Roy's nth birthday can be expressed as:
\[
f(n) = 100 + (n - 1) \cdot 80
\]
Thus, the correct model for the situation is:
\[
f(n) = 100 + (n - 1) \cdot 80
\]
In conclusion, the correct response is:
**f(n) = 100 + (n - 1)80**
1. At \( n = 1 \) (Roy's first birthday), the balance is $100.
2. At \( n = 2 \) (Roy's second birthday), the balance is \( 100 + 80 = 180 \).
3. At \( n = 3 \) (Roy's third birthday), the balance is \( 100 + 80 + 80 = 260 \).
4. At \( n = 4 \) (Roy's fourth birthday), the balance is \( 100 + 80 + 80 + 80 = 340 \).
From the pattern, we observe that with each passing year, Roy's account balance increases by $80. The balance can be expressed as follows:
- For \( n = 1 \): \( 100 + 0 \cdot 80 = 100 \)
- For \( n = 2 \): \( 100 + 1 \cdot 80 = 180 \)
- For \( n = 3 \): \( 100 + 2 \cdot 80 = 260 \)
- For \( n = 4 \): \( 100 + 3 \cdot 80 = 340 \)
The general formula for the balance on Roy's nth birthday can be expressed as:
\[
f(n) = 100 + (n - 1) \cdot 80
\]
Thus, the correct model for the situation is:
\[
f(n) = 100 + (n - 1) \cdot 80
\]
In conclusion, the correct response is:
**f(n) = 100 + (n - 1)80**