Question

Hey, i need help with this question. it is attached below.

Answer 1

(b)$51,571

Explanation:

Rhonda's profit in first year = $20,000

Her annual profit increased by 7% each year.

(a)Iterative Rule

An iterative rule to model the sequence formed by the profits of Rhonda's business each year is:

`\begin{gathered} P(n)=(1+7\%)P_(n-1)=(1+0.07)P_(n-1) \\ \implies P(n)=1.07P_(n-1),P(1)=20,000 \end{gathered}`

The iterative rule is:

`\begin{gathered} P(n)=1.07P_(n-1),n\geq2 \\ P\left(1\right)=20,000 \end{gathered}`

(b)

Using the rule, we want to find the value of P(15), the business' predicted profit 15 years from the start.

`\begin{gathered} P(2)=1.07* P_(2-1)=1.07* P_1=1.07*20,000 \\ P(3)=1.07* P_(3-1)=1.07* P_2=1.07*(1.07*20000)=1.07^2*20,000 \\ \implies \\ P(15)=1.07^(15-1)*20,000 \\ =1.07^(14)*20,000 \\ =51570.68 \\ \approx51,571 \end{gathered}`

The business' predicted profit 15 years from the start is approximately $51,571 (rounded to the nearest dollar).