a) after how many years will she have 64 rabbits?

b) write and equation to model this situation

Show work

It is geometric sequence.

Therefore:

\( a_1=4\\ q=2\\ S=a_1*\frac{1-q^n}{1-q}\\ 64=4*\frac{1-2^n}{1-2}\\ 16=\frac{1-2^n}{-1}\\ -16=1-2^n\\ 16+1=2^n\\ 2^4+1=2^n\\ n>4 \)

After 4 yours she will have over 64 rabbits.

\( a)\\n-after\ how\ many\ years\ will\ she\ have\ 64\ rabbits\\ \\4\cdot \frac{1-2^n}{1-2} =64\ /:4\\ \\\frac{1-2^n}{-1} =16\ \ \ \Leftrightarrow\ \ \ 2^n-1=16\ \ \ \Leftrightarrow\ \ \ 2^n=17>16\\ \\2^n>2^4\\.\ \ \ \ \ \ \ \ \ \ \ \ \ \ a^b> a^c\ \ \wedge\ \ \ a>1\ \ \ \Rightarrow\ \ \ b> c\\n>4\\ \\Ans. \ Yolanda\ have\ 64\ rabbits\ after\ 4\ years \)

\( b)\\b-how\ much\ rabbits\ was\ in\ the\ beginning\\t-how\ many\ times\ will\ be\ an\ increase\ in\ the\ number\ of\ rabbits\\.\ \ \ \ in\ the\ years\\e-how\ many\ rabbits\ are\ expected\ Yolanda\\ \\ b\cdot \frac{1-t^n}{1-t} =e \)