The number of bacteria in a culture is 1000 and this number increases by 250% every two hours. How many bacteria is present after 24 hours?

Question

Had to find a recurrent relation for this problem and then find the explicit formula to get our bacteria amount after 24 hours.

I got the question wrong because my recurrence relation was wrong. Trying to get some more insight as to why it is wrong as the teacher didn't give reasons.

What I did:

The initial condition is $a_0=1000$

The percent increase is 2.5 per 2 hours

$a_n$= number of two-hour blocks (12 in 24 hour period): $1\le n\le12$

$a_1=a_0(2.5) + a_0 = a_0(2.5 + 1) = a_0(3.5)$

thus we get $a_n=3.5a_{n-1}$ for the relation

with an explicit function $a_n = 3.5^n(1000)$

However I was corrected to $a_n=2.5a_{n-1}$

and $a_n=2.5^n(1000)$

Why is it 2.5 rather than 3.5? I interpreted the question as every two-hour block $a_n$ will increase 250% so when we work out $a_n$ we would need to find the percent amount for $a_{n-1}$ added to the original amount of $a_{n-1}$. So each $a_n$ would need $a_{n-1}(2.5) + a_{n-1}$ as I did above where factoring the $a_{n-1}$ yields $a_n=(3.5)a_{n-1}$

When I check the increase per $a_n$ and $a_{n+1}$ with the $a_n=3.5a_{n-1}$ relation we get the 250% increase per 2-hour block; using $a_n=2.5a_{n-1}$ relation is 150% increase per 2-hour block.

$a_1 = 3.5a_0 = 3500$

$a_2 = 3.5a_1 = 12250$

$(12250 - 3500)/3500 = 2.5(100) = 250$% increase

$a_1=2.5a_0 = 2500$

$a_2=2.5a_1 = 6250$

$(6250-2500)/2500 = 1.5(100) = 150$% increase

So my autistic brain is missing something. Please Help

Answer 1

$250\%$ incresse means the increase is $250\%$.

So if we start off with $1000$ bacteria, and say after $2$ hours the population of bacteria = $b$ $b = 1000 + 250\% \times 1000 = 1 \times 1000 + 250\% \times 1000$
$b = 100\% \times 1000 + 250\% \times 1000$ (because $1 = 100\%$)
$b = (100\% + 250\%) \times 100$
$b = 350\% \times 1000$

$350\% = 3.5$

The population of bacteria at hour $h$ is $b_h$
$b_h = 3.5^{\frac{h}{2}} \times 1000$

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It depends on what you're counting. If you're counting ...

1. The population then $3.5^{\frac{h}{2}} \times 1000$ after $h$ hours
2. The increase in population then $2.5^{\frac{h}{2}} \times p$ where $p$ is the population in the previous hour.

$b_3 = ((1000(1 + 2.5))(1 + 2.5))(1 + 2.5)$. When expressed this way you can see what's actually going on. You're taking the previous 2=hour's population ($1$), calculating the increase in population ($2.5$) and then adding ($1 + 2.5)$ to get the new population.