Question
A college fund is started with an initial deposit of $2,000.00. The amount in the fund after x number of years is represented by the recursive equation f(x)=f(x−1)⋅1.06
, in which f(0)=2,000
. Use this recursive formula to create an exponential equation that models the fund’s balance, A
, after t years, assuming there were no additional deposits or withdrawals. Which of the following is the correct equation?(1 point)
Responses
A=2,000e0.061t
upper A equals 2,000 e superscript 0.061 t baseline
A=2,000⋅1.06t
upper A equals 2,000 times 1.06 superscript t baseline
A=2,000e0.058t
upper A equals 2,000 e superscript 0.058 t baseline
A=2,000e0.06t
, in which f(0)=2,000
. Use this recursive formula to create an exponential equation that models the fund’s balance, A
, after t years, assuming there were no additional deposits or withdrawals. Which of the following is the correct equation?(1 point)
Responses
A=2,000e0.061t
upper A equals 2,000 e superscript 0.061 t baseline
A=2,000⋅1.06t
upper A equals 2,000 times 1.06 superscript t baseline
A=2,000e0.058t
upper A equals 2,000 e superscript 0.058 t baseline
A=2,000e0.06t
Answers
GPT-4o mini
The recursive formula given is:
\[ f(x) = f(x-1) \cdot 1.06 \]
with the initial condition:
\[ f(0) = 2000 \]
This means that the amount in the fund increases by 6% each year.
To derive the explicit formula from the recursive relationship, we can rewrite the recursive formula in terms of \( t \) (where \( t \) represents the number of years) as follows:
- For \( t = 1 \):
\[ f(1) = f(0) \cdot 1.06 = 2000 \cdot 1.06 \]
- For \( t = 2 \):
\[ f(2) = f(1) \cdot 1.06 = (2000 \cdot 1.06) \cdot 1.06 = 2000 \cdot (1.06)^2 \]
- Continuing this pattern, for \( t \):
\[ f(t) = 2000 \cdot (1.06)^t \]
Thus, the formula for the amount \( A \) in the fund after \( t \) years can be expressed as:
\[ A = 2000 \cdot (1.06)^t \]
This equation represents an exponential growth model based on an initial amount of $2,000 and a growth factor of 1.06 for each year.
Therefore, the correct equation is:
\[ A = 2000 \cdot 1.06^t \]
This matches the second option:
**A = 2,000 ⋅ 1.06^t**
\[ f(x) = f(x-1) \cdot 1.06 \]
with the initial condition:
\[ f(0) = 2000 \]
This means that the amount in the fund increases by 6% each year.
To derive the explicit formula from the recursive relationship, we can rewrite the recursive formula in terms of \( t \) (where \( t \) represents the number of years) as follows:
- For \( t = 1 \):
\[ f(1) = f(0) \cdot 1.06 = 2000 \cdot 1.06 \]
- For \( t = 2 \):
\[ f(2) = f(1) \cdot 1.06 = (2000 \cdot 1.06) \cdot 1.06 = 2000 \cdot (1.06)^2 \]
- Continuing this pattern, for \( t \):
\[ f(t) = 2000 \cdot (1.06)^t \]
Thus, the formula for the amount \( A \) in the fund after \( t \) years can be expressed as:
\[ A = 2000 \cdot (1.06)^t \]
This equation represents an exponential growth model based on an initial amount of $2,000 and a growth factor of 1.06 for each year.
Therefore, the correct equation is:
\[ A = 2000 \cdot 1.06^t \]
This matches the second option:
**A = 2,000 ⋅ 1.06^t**