Thesleepystudent
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Table shows the number of bacteria present in a particular sample for the first 5 minutes

For one strain of bacteria, each bacterium divides into two every minute.

Time Bacteria present
0 2
1 4
2 8
3 16
4 32
5 64
1)write down an algebraic rule linking the number of bacteria present at a particular time to the number present one minute previously.
(This I can do it's a geometric formula)

2) Write down an expression for the number of bacteria present after t minutes.

3) calculate the number of bacteria after 2 hours.(state any assumptions you can make)

4) calculate the time it takes for the colony to reach 1 million bacteria. ( Does this require logs and sum of series).

Any help would mean a lot.
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mqb2766
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(Original post by Thesleepystudent)
Table shows the number of bacteria present in a particular sample for the first 5 minutes

For one strain of bacteria, each bacterium divides into two every minute.

Time Bacteria present
0 2
1 4
2 8
3 16
4 32
5 64
1)write down an algebraic rule linking the number of bacteria present at a particular time to the number present one minute previously.
(This I can do it's a geometric formula)

2) Write down an expression for the number of bacteria present after t minutes.

Any help would mean a lot
Instead of a term to term geometric relationship (part 1), what do you know about position to term geometric relationship? That's what you need for part 2.
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Sinnoh
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If you just wrote down the powers of 2, up to 64 (26), it might become obvious what the general rule is
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Thesleepystudent
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(Original post by mqb2766)
Instead of a term to term geometric relationship (part 1), what do you know about position to term geometric relationship? That's what you need for part 2.
Should I rearrange the equation to make t the subject so it is in the (Un= ) postion
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Thesleepystudent
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(Original post by Sinnoh)
If you just wrote down the powers of 2, up to 64 (26), it might become obvious what the general rule is
I understand the general rule is a geometric sequence however I'm unsure about question 2
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Sinnoh
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(Original post by Thesleepystudent)
I understand the general rule is a geometric sequence however I'm unsure about question 2
All it's asking for is to write a function that tells you the number of bacteria at a given time. It'll have to be an exponential, and there would have to be a 2 as the basis of the exponent since it doubles at each minute.

(Original post by Thesleepystudent)
Should I rearrange the equation to make t the subject so it is in the (Un= ) postion
That would give you the time elapsed for a given number of bacteria - they're asking for the opposite of that.
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mqb2766
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(Original post by Thesleepystudent)
Should I rearrange the equation to make t the subject so it is in the (Un= ) postion
If it was an arithmetic sequence you'd have something like
Un = 5 + 3(n-1)
Where 5 is the initial value and 3 the common difference.

For a geometric sequence it's
Un = a*r^(n-1)
So identify the initial value "a" and the common ratio "r"
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Thesleepystudent
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(Original post by mqb2766)
If it was an arithmetic sequence you'd have something like
Un = 5 + 3(n-1)
Where 5 is the initial value and 3 the common difference.

For a geometric sequence it's
Un = a*r^(n-1)
So identify the initial value "a" and the common ratio "r"
a=2 and r=2. n becomes t for the first half. However how could I find after t for the second half.

(Original post by Sinnoh)
All it's asking for is to write a function that tells you the number of bacteria at a given time. It'll have to be an exponential, and there would have to be a 2 as the basis of the exponent since it doubles at each minute.



That would give you the time elapsed for a given number of bacteria - they're asking for the opposite of that.
Thank you for your help so after t minute Un= 2^n+1 because each power goes up by 1 so if n =1 the answer will be 4.
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Sinnoh
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(Original post by Thesleepystudent)
a=2 and r=2. n becomes t for the first half. However how could I find after t for the second half.


Thank you for your help so after t minute Un= 2^n+1 because each power goes up by 1 so if n =1 the answer will be 4.
They want it in terms of t. Like, if N is the number of bacteria, then N = f(t).
But you're very close to the right answer.
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Thesleepystudent
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(Original post by Sinnoh)
They want it in terms of t. Like, if N is the number of bacteria, then N = f(t).
But you're very close to the right answer.
Thank you for the help could you check I'm on the right track. I feel like I'm a little off.
So far for part 1) 2*2^t-1= 2^t-1 we replace n with t because we're looking for the time(t) at a particular time.
Part 2) Ut= 2^t+1 the base will be 2 as the number of bacteria doubles each time with power going up by 1 each time.
And finally part 3) I just put 2 hours(120 seconds) into the equation in part 2 and not quite sure what I make assumptions on.
Thanks once again,
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Sinnoh
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(Original post by Thesleepystudent)
Thank you for the help could you check I'm on the right track. I feel like I'm a little off.
So far for part 1) 2*2^t-1= 2^t-1 we replace n with t because we're looking for the time(t) at a particular time.
Part 2) Ut= 2^t+1 the base will be 2 as the number of bacteria doubles each time with power going up by 1 each time.
And finally part 3) I just put 2 hours(120 seconds) into the equation in part 2 and not quite sure what I make assumptions on.
Thanks once again,
Yeah that's correct so far.
The main assumption is that the number of bacteria grow without limit, continuing to double every minute - in practice they'd run out of space or food or die off. 2^119 is a pretty big number after all. That much bacteria would probably weigh as much as the Moon.

The question describes it as "bacteria present", so for 4) you don't need to interpret it as the sum of a series.
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