Solve each problem. If certain bacteria are cultured in a...

Solve each problem. If certain bacteria are cultured in a medium with sufficient nutrients, they will double in size and then divide every 40 minutes. Let $N_{1}$ be the initial number of bacteria cells, $N_{2}$ the number after 40 minutes, $N_{3}$ the number after 80 minutes, and $N_{j}$ the number after $40(j-1)$ minutes. (a) Write $N_{j+1}$ in terms of $N_{j}$ for $j \geq 1$ (b) Determine the number of bacteria after 2 hr if $N_{1}=230$. (c) Graph the sequence $N_{j}$ for $j=1,2,3, \ldots, 7,$ where $N_{1}=230 .$ Use the window $[0,10]$ by $[0,15,000]$ (d) Describe the growth of these bacteria when there are unlimited nutrients.

Solve each problem.
If certain bacteria are cultured in a medium with sufficient nutrients, they will double in size and then divide every 40 minutes. Let $N_{1}$ be the initial number of bacteria cells, $N_{2}$ the number after 40 minutes, $N_{3}$ the number after 80 minutes, and $N_{j}$ the number after $40(j-1)$ minutes.
(a) Write $N_{j+1}$ in terms of $N_{j}$ for $j \geq 1$
(b) Determine the number of bacteria after 2 hr if $N_{1}=230$.
(c) Graph the sequence $N_{j}$ for $j=1,2,3, \ldots, 7,$ where $N_{1}=230 .$ Use the window $[0,10]$ by $[0,15,000]$
(d) Describe the growth of these bacteria when there are unlimited nutrients.

Key Concepts

Graphing Discrete Sequences

Graphing discrete sequences involves plotting individual values on a coordinate system to visualize trends and patterns. This technique helps in understanding the behavior of sequences, such as the rapid increases seen in exponential growth, by representing each term as a distinct point.

Exponential Growth

Exponential growth occurs when a quantity increases by a constant multiplicative rate over equal time intervals. This type of growth is typical in populations or processes where resources are unlimited, leading to rapid increases over time.

Geometric Sequence

A geometric sequence is one where the ratio between consecutive terms is constant. This concept is critical when each term is a fixed multiple of the previous term, leading to an explicit formula that describes the sequence's behavior.

Recurrence Relation

A recurrence relation defines each term of a sequence as a function of its preceding term(s). For example, writing N????? in terms of N? models how a quantity evolves step by step over time.

Transcript

00:01 Hello, hope you're doing well.

00:03 So for this problem, we're told that at n1 is the number of bacteria at the beginning.

00:12 And then n2, it's going to be the number of bacteria after 40 minutes.

00:18 Then n3 is going to be the number of bacteria after another 40 minutes and so on.

00:23 The problem tells us that for every 40 minutes, the bacteria double their size and then divide.

00:30 So basically that means that from n1 to n2, the number of bacteria is going to double.

00:37 And same from n2 to n3 and et cetera.

00:41 So that's what the problem tells us.

00:42 So the first part of the problem asks us to figure out if we're given n, so given nj plus 1 and nj, what's the formula or the equation that relates these two? how do we get from nj to nj plus one? and then the second part of the problem, we need to use that formula to say, okay, after two hours, given a certain n1 value, what's the number of bacteria you're going to be? and then last, we need to graph it and kind of see the trend that the number of bacteria, like how it increases over time.

01:20 So starting with part a, i mean to figure out the formula for nj plus 1 in terms of nj.

01:27 So you can see here that, again, like i said before, the problem tells us that every 40 minutes, so between n1 and n2, between n2 and n3, et cetera, the bacteria double in size and then divide.

01:40 So n2 is equal to two times the number of bacteria at n1.

01:45 And then again, the same thing happens over the next 40 minutes.

01:49 At n3, the number of bacteria 40 minutes later is double the number of bacteria at n1.

01:56 2.

01:57 So this means that for any nj plus 1 number of bacteria is double the number of bacteria at the previous 40 minutes.

02:07 It's equal to 2 times nj.

02:11 And this is our answer right here for part a of this problem.

02:16 So now we need to figure out how many bacteria there are after two hours.

02:22 So since there are 60 minutes in an hour, this is equal to 120 minutes.

02:27 So we're told that n1 is equal to 230.

02:33 So we know that nj plus 1 is equal to two times nj, so that means that n2 is going to be double n1, which is equal to two times 230, which is 460.

02:45 So this is after 40 minutes.

02:50 So now moving on to n3, n3 is at 80 minutes, so 40 minutes later is equal to two times n2, which is 2 times 460, which is 920.

03:03 So this is at 80 minutes.

03:05 Lastly, we have n4, which is 40 minutes later...

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