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+(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
f left parenthesis n right parenthesis equals 80 left parenthesis 100 right parenthesis superscript n minus 1 baseline
f(n)=100+80(n)
Responses
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
f left parenthesis n right parenthesis equals 80 left parenthesis 100 right parenthesis superscript n minus 1 baseline
f(n)=100+80(n)
Answers
GPT-4o mini
To model the situation described, let's analyze the contributions to the bank account over the years.
1. On Roy's first birthday, his grandfather deposits $100, giving an initial amount of $100.
2. On each subsequent birthday (from the second to the nth birthday), he deposits an additional $80.
The balance on each birthday can be expressed as follows:
- After the first birthday (n=1):
\[ f(1) = 100 \]
- After the second birthday (n=2):
\[ f(2) = 100 + 80 = 180 \]
- After the third birthday (n=3):
\[ f(3) = 100 + 80 \times 2 = 260 \]
- After the fourth birthday (n=4):
\[ f(4) = 100 + 80 \times 3 = 340 \]
From this, we can see a pattern: the initial deposit of $100 remains constant, and each subsequent year, $80 is added for every birthday after the first.
To generalize this, the function can be expressed as:
\[
f(n) = 100 + 80(n-1)
\]
Therefore, the correct function that models this situation is:
\[
f(n) = 100 + 80(n-1)
\]
So the appropriate option from the given choices is:
\[
f(n) = 100 + (n-1)80
\]
1. On Roy's first birthday, his grandfather deposits $100, giving an initial amount of $100.
2. On each subsequent birthday (from the second to the nth birthday), he deposits an additional $80.
The balance on each birthday can be expressed as follows:
- After the first birthday (n=1):
\[ f(1) = 100 \]
- After the second birthday (n=2):
\[ f(2) = 100 + 80 = 180 \]
- After the third birthday (n=3):
\[ f(3) = 100 + 80 \times 2 = 260 \]
- After the fourth birthday (n=4):
\[ f(4) = 100 + 80 \times 3 = 340 \]
From this, we can see a pattern: the initial deposit of $100 remains constant, and each subsequent year, $80 is added for every birthday after the first.
To generalize this, the function can be expressed as:
\[
f(n) = 100 + 80(n-1)
\]
Therefore, the correct function that models this situation is:
\[
f(n) = 100 + 80(n-1)
\]
So the appropriate option from the given choices is:
\[
f(n) = 100 + (n-1)80
\]