Yolanda owns 4 rabbits. She expects the number of rabbits to double every year.

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.

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


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


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