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

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=2,000e0.06t
upper A equals 2,000 e superscript 0.06 t baseline

A=2,000e0.058t
upper A equals 2,000 e superscript 0.058 t baseline

A=2,000e0.061t
upper A equals 2,000 e superscript 0.061 t baseline

A=2,000⋅1.06t

GPT-4o mini · Answer

The recursive formula you provided is: $$ f(x) = f(x-1) \cdot 1.06 $$ with the initial condition $ f(0) = 2000 $. This indicates that the amount in the fund grows by a factor of $ 1.06 $ each year. Therefore, if we want to express this in a closed-form exponential equation, we can analyze the recursive relationship. Starting with the base case: - For $ x = 0 $: $$ f(0) = 2000 $$ - For $ x = 1 $: $$ f(1) = f(0) \cdot 1.06 = 2000 \cdot 1.06 $$ - For $ x = 2 $: $$ f(2) = f(1) \cdot 1.06 = (2000 \cdot 1.06) \cdot 1.06 = 2000 \cdot (1.06)^2 $$ - For $ x = t $: $$ f(t) = 2000 \cdot (1.06)^t $$ This leads us to the closed-form equation, which represents the amount in the fund after $ t $ years as: $$ A = 2000 \cdot 1.06^t $$ Among the provided options, the correct equation is: $$ A = 2000 \cdot 1.06^t $$