Question

A population grows according to an exponential growth model. The initial population is P0=9, and the growth rate isr=0.4.Then:P1 = P2 = Find an explicit formula for Pn. Your formula should involve NPn = Use your formula to find P9=Give all answers accurate to at least one decimal place

Answer 1

From the question,

`\begin{gathered} P_0=9 \\ r=0.4 \end{gathered}`

The formula for the growth rate will be calculated using the formula below

`\begin{gathered} F=P(1+r)^n \\ F=\text{future value} \\ P=present\text{ value} \\ r=\text{growth rate} \\ n=nu\text{mber of times per period} \end{gathered}`

In,

`\begin{gathered} P_0,n=0 \\ P_1,n=1 \\ P_2,n=2 \\ P_9,n=9 \end{gathered}`

Given that

`\begin{gathered} P_0=9 \\ F=P(1+r)^n \\ P_0=9(1+0.4)^0 \\ P_0=9*1 \\ P_0=9 \end{gathered}`
`\begin{gathered} F=P(1+r)^n \\ P_1=9(1+0.4)^1 \\ P_1=9*1.4 \\ P_1=12.6 \end{gathered}`

Hence,

P1 = 12.6

Also, we will have P2 to be

`\begin{gathered} F=P(1+r)^n \\ P_2=9(1+0.4)^2 \\ P_2=9*1.4^2 \\ P_2=17.64 \\ P_2\approx1\text{ decimal place} \\ P_2=17.6 \end{gathered}`

Hence,

P2 = 17.6

Therefore,

The formula for Pn will be represented below as

`\begin{gathered} P_n=9(1+0.4)^n \\ P_n=9(1.4)^n \end{gathered}`

The explicit formula for Pn will be

`P_n=9(1.4)^n`

To figure out the values of P9. we will substitute the value of n=9 in the equation below

`\begin{gathered} P_n=9(1.4)^n \\ P_9=9(1.4)^9 \\ P_9=185.9 \end{gathered}`

Hence,

P9 = 185.9