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In the beginning there are about 1000 bacteria in a testing tube and the amount of bacteria increases every two hours with 250 %. Use a recursion ecvation to tell the amount of bacteria after n days?

I made a formula for every two hours but i dont know how to countine after that.

y = (2,5x/2) + 1000

Where x is the amount of hours while y is the total amount of bacteria after x hours.

Edit: It should increase by 250% of what it was two hours ago, should be correct for those who noted my problems

Varrens
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  • This is unfortunately incorrect. Let $f(x)$ represent the function denoting how many bacteria there are after $x$ hours. We are told that $f(0)=1000$. We are told also that after two hours the amount of bacteria that was there is increased to $250%$ of what it was previously (or maybe you mean increased /by/ $250%$ instead) which would make $f(2)=2.5\cdot 1000=2.5\cdot f(0)$. At the time $x=4$, it will have again increased to $250%$ of what it was two hours earlier, i.e. $f(4)=2.5\cdot f(2)=2.5\cdot (2.5\cdot f(0))=2.5\cdot (2.5\cdot 1000)$ etc... – JMoravitz May 28 '18 at 18:03
  • Now... see if you can generalize. Keep in mind when you should be using addition, when you should be using multiplication, when you should be using division, and when you should be using exponentiation. (Also, you should check whether you intend that it increases \to\ $250%$ of the amount two hours prior, or if you intend that it increases \by\ $250%$ of the amount two hours prior. These mean different things and will lead to different final answers. In the first interpretation, after two hours from the start there are $2500$ bacteria, in the second there are $3500$ instead) – JMoravitz May 28 '18 at 18:04

1 Answers1

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The recursion formula is given by

$B_{n+1} = B_n+\frac{250}{100}B_n = \frac{350}{100}B_n$

We can rewrite this as;

$B_n = 3.5\cdot B_{n-1} =3.5(3.5\cdot B_{n-2})=3.5(3.5(3.5\cdot B_{n-3}))=\cdots=(3.5)^n\cdot B_0$

so as ; $B_0 = 1000$

$B_1 = 3.5\times 1000 = 3500$

$\vdots$

$B_n = (3.5)^n\cdot B_0$ where $n$ is the $n^{th}$ two hour mark.

The Integrator
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  • Two problems: the reported $250%$ increase occurs every two hours, not every one hour. Second, it has not yet been clarified what the phrase "the amount of bacteria increases every two hours with 250 %" means, whether it means that it should increase by $250%$ of what it was two hours ago or if it should increase to $250%$ of what it was two hours ago. – JMoravitz May 28 '18 at 18:13
  • @JMoravitz i am not sure how the first problem arose from my answer but i totally agree with second one. I'll wait for an update from OP and try to fix it. If i am wrong in any way I'd be glad if you could help me rectify it. Thank you:) – The Integrator May 28 '18 at 18:15