Question

Help I need help with this question, and your help would be appreciated very much, as it's helpful.

Answer 1

Before we create the equation let's see how it grows, suppose that he starts with 8 villagers, so the first town has 8 villagers and the second town will have 1.17 more, then

`\begin{gathered} 1\Rightarrow8 \\ 2\Rightarrow8\cdot1.17=9.36 \end{gathered}`

So at the second village, he can have 9.36 villagers, let's round it to 9 villagers, at the third village he can have 1.17 more, but not from 8 villagers, from 9.36 villagers, then

`\begin{gathered} 1\Rightarrow8 \\ 2\Rightarrow8\cdot1.17=9.36 \\ 3\Rightarrow9.36\cdot1.17=10.9512 \end{gathered}`

So at the third village, he can have 10.95 villagers, but let's rewrite it to see something interesting, instead of 9.36 let's use 8*1.17, it will help us to create our equation

`\begin{gathered} 1\Rightarrow8 \\ 2\Rightarrow8\cdot1.17 \\ 3\Rightarrow8\cdot1.17\cdot1.17 \end{gathered}`

Look that we have 1.17 multiplied two times, so

`\begin{gathered} 1\Rightarrow8 \\ 2\Rightarrow8\cdot1.17 \\ 3\Rightarrow8\cdot1.17^2 \end{gathered}`

Now let's see the fourth village, it will be the same as the third multiplied by 1.17 again, then

`\begin{gathered} 1\Rightarrow8 \\ 2\Rightarrow8\cdot1.17 \\ 3\Rightarrow8\cdot1.17^2 \\ 4\Rightarrow8\cdot1.17^3 \end{gathered}`

As we can see, we're doing an exponential growing, we can already guess that at the fifth village it will be

`5\Rightarrow8\cdot1.17^4`

Now we already have the pattern, the generic way to express it will be

`8\cdot1.17^(n-1)`

Where "n" is the village, but why (n-1) at the exponential? that's because at the first village (n = 1), we will have

`1\Rightarrow8\cdot1.17^(n-1)=8\cdot1.17^0=8^{}`

As expected, so we can say that our equation is

`V=8\cdot1.17^((n-1)),n=1,2,3,4\ldots`

Then to find out the population at the 16th village, let's put n = 16, then

`V=8\cdot1.17^((16-1))=8\cdot1.17^(15)=84.30977`

We can round it to 84 villagers.