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)=100+80(n) f left parenthesis n right parenthesis equals 100 plus 80 left parenthesis n right parenthesis 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+(n−1)80 f left parenthesis n right parenthesis equals 100 plus left parenthesis n minus 1 right parenthesis 80 f(n)=80(100)n−1
Answers
GPT-4o mini
To create a function that models the situation described, we need to account for the initial deposit of $100 on Roy's first birthday and the $80 increments that follow for each year thereafter.
Given:
- On Roy's first birthday (n=1), the amount in the account is $100.
- On Roy's second birthday (n=2), the account balance becomes $100 + $80 = $180.
- On the third birthday (n=3), the balance is $100 + $80 + $80 = $260.
- On the fourth birthday (n=4), the balance is $100 + $80 + $80 + $80 = $340.
The pattern shows that the balance can be modeled with a function that includes the initial deposit and the linear accumulation of $80 for each passing year after the first.
The balance function can be represented as:
- For the first year (n=1), \( f(1) = 100 \)
- For the second year (n=2), \( f(2) = 100 + 80 \times 1 = 180 \)
- For the third year (n=3), \( f(3) = 100 + 80 \times 2 = 260 \)
- For the fourth year (n=4), \( f(4) = 100 + 80 \times 3 = 340 \)
We can generalize this into a function:
\[
f(n) = 100 + 80(n - 1)
\]
This can also be written equivalently as:
\[
f(n) = 100 + 80n - 80 = 80n + 20
\]
However, the first option provided in your responses matches perfectly with the sequence for the years after the first year:
\[
f(n) = 100 + 80(n - 1)
\]
So, the correct function that models this situation is:
\[
f(n) = 100 + 80(n - 1)
\]
This is the response you would select from the options you've given.
Given:
- On Roy's first birthday (n=1), the amount in the account is $100.
- On Roy's second birthday (n=2), the account balance becomes $100 + $80 = $180.
- On the third birthday (n=3), the balance is $100 + $80 + $80 = $260.
- On the fourth birthday (n=4), the balance is $100 + $80 + $80 + $80 = $340.
The pattern shows that the balance can be modeled with a function that includes the initial deposit and the linear accumulation of $80 for each passing year after the first.
The balance function can be represented as:
- For the first year (n=1), \( f(1) = 100 \)
- For the second year (n=2), \( f(2) = 100 + 80 \times 1 = 180 \)
- For the third year (n=3), \( f(3) = 100 + 80 \times 2 = 260 \)
- For the fourth year (n=4), \( f(4) = 100 + 80 \times 3 = 340 \)
We can generalize this into a function:
\[
f(n) = 100 + 80(n - 1)
\]
This can also be written equivalently as:
\[
f(n) = 100 + 80n - 80 = 80n + 20
\]
However, the first option provided in your responses matches perfectly with the sequence for the years after the first year:
\[
f(n) = 100 + 80(n - 1)
\]
So, the correct function that models this situation is:
\[
f(n) = 100 + 80(n - 1)
\]
This is the response you would select from the options you've given.