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I am trying to determine the following. A bacteria culture starts with 500 bacteria and doubles its size every hour. This means that the population of the bacteria at the end of each hour will be 2 times that at the end of the previous hour.

If I let t be the number of hours after the culture starts, and $N(t)$ the population of the bacteria at the end of the t-th hour. Then I get $N(t)=500⋅2t.$

Does this seem correct?

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    $\begingroup$ With you formula, at $N(2)=2000$, but $N(3)=3000$ which means the culture didn't double in size in that hour. Also, your formula give $N(0)=0$, when we know it should be $500$. For your formula, you will need something that will multiply $500$ by $2$ again every hour. What mathematical operation will multiply by $2$ a set number of times? $\endgroup$
    – Tyberius
    Jul 9, 2021 at 19:56

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No, it is incorrect. Let me explain why.

Let's test out a few values of your function and compare them with the correct values:

After 1 hour since the population starts, the bacteria should be at $500 \cdot 2=1,000$, and your function gives $500 \cdot 2 \cdot (1) = 1,000$, so after one hour your function gives the correct result.

After 2 hours since the population starts, the bacteria should be at $500 \cdot 2 \cdot 2=2,000$, and your function gives $500 \cdot 2 \cdot (2) = 2,000$, so after two hours your function gives the correct result.

After 3 hours since the population starts, the bacteria should be at $500 \cdot 2 \cdot 2 \cdot 2 = 4,000$, and your function gives $500 \cdot 2 \cdot (3) = 3,000$, so after three hours your function gives a wrong result.



Hmmm... looking at the correct population, I seem to notice a pattern. Because we are doubling the previous population every time, we can see that, for instance, after the first hour we have one 2, after the second hour we have two 2's, etc.

Using this, we can build our equation. We have $N(t)=500*(\text{as many 2's as t's})$. What function can we use such that we multiply $2$ by itself $t$ times? Exponentiation!

$2^{t}$, for a positive integer $t$ is the same as multiplying $2$ by itself $t$ times.


Therefore, our final equation is $\boxed{N(t)=500*2^{t}}$

I'll leave it as an exercise to test a couple values of $N(t)$ to make sure it works.

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  • $\begingroup$ $(+1)$. I suggest that you leave out the stuff on pi notation, because I find it unlikely that the OP would be familiar with it. As you have already said, in this case, $\Pi_{i=1}^{t}2$ just means "multiply $2$ by itself $t$ times", so there's no need for fancy notation. But you don't have to take my advice. $\endgroup$
    – Joe
    Jul 9, 2021 at 21:29
  • $\begingroup$ Ok, sure. Thanks for the help. $\endgroup$
    – dgeyfman
    Jul 9, 2021 at 21:30
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You have to stretch it even further.

You assumed a multiplicative relationship $ 500 *2t$,but you wanted exponential growth ( base is different to $e$ )... which is repeated multiplication

$$ 500 *2^t$$

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