Question

A store's sales grow according to the recursive rule Pn = Pn-1 + 10000, with initial sales Po = 26000.(a) Calculate P and P2.P = $P2 = $(b) Find an explicit formula for Pr.Pn -(c) Use the explicit formula to predict the store's sales in 10 years.Plo - S(d) When will the store's sales exceed $139,000? Round your answer to the nearest tenth of a year.Afteryears.

Answer 1

Given the following recursive rule:

`\begin{gathered} P_n=P_(n-1)_{}+10000 \\ P_o=26000 \end{gathered}`

We will find the following:

(a) Calculate P₁ and P₂

so, the value of P₁ = 26000

and P₂ = P₁ + 10000 = 26000 + 10000 = 36000

P₂ = 36000

(b) Find an explicit formula for Pn

so,

`\begin{gathered} P_n=P_o+d(n-1) \\ P_n=26000+10000(n-1) \end{gathered}`

(c) Use the explicit formula to predict the store's sales in 10 years.

so, substitute with n = 10

`\begin{gathered} P_(10)=26000+10000\cdot(10-1)=26000+10000\cdot9 \\ \\ P_(10)=116000 \end{gathered}`

(d) When will the store's sales exceed $139,000?

so, we will substitute with Pn = 139000, then solve the equation to find (n)

`\begin{gathered} 139000=26000+10000(n-1) \\ 10000(n-1)=139000-26000 \\ 10000(n-1)=113000 \\ n-1=(113000)/(10000)=11.3 \\ n=11.3+1=12.3 \end{gathered}`

So, the answer will be after 12.3 years