 Let's take a look at geometric growth in plants. Before we get into plants, let's talk a little bit about bacteria because when bacteria grow and multiply, they do follow geometric growth. Let's say we have one bacterium which can divide. And when these two cells divide, they form four bacteria. And then again, eight bacteria. So if you notice, every time the bacteria are dividing, the number of cells added is increasing. So we start off with one cell, right? And then how many are added? The first time it divides one. So we add one bacterium to get two cells. And then now how many are added? Two cells to form four cells. And then now four cells are added. So each time the bacteria are dividing, the number of cells added is going on increasing. So this is typically what happens in geometric growth. So let's try to quantify all this. Let's say we have 100 bacteria in this Petri plate. And let's say in one hour, two bacteria are added, right? So after one day, how many bacteria will be there? So how do we calculate this? So first of all, let's see what's happening after one hour. After one hour, two bacteria are added. And so how many total bacteria are there? We started out with 100 bacteria. And then each of them divide to form two bacteria. So we multiply 100 by 2. So 100 by 2 is the number of bacteria that are added. So let's write this as 100 times 2 to the power 1. Why we are writing like this? It will be clear in a moment. And this is equal to 200, of course. Now what happens in two hours? In two hours, so already we had 200 bacteria. And we multiply it by 2. We can also write this as 100 times 2 to the power 2. So that's 400. Similarly, for three hours we could write number of bacteria that are there is 400 times 2 is equal to 100 times 2 to the power 3, which is 800. So do you see a pattern? You see whatever the number of hours, if you just raise 2 to the power of that number and then multiply by 100, you get the total number of bacteria that formed after that many hours. So then after one day, how many bacteria are formed? One day is 24 hours. So following this pattern, it's clear that after 24 hours, it will be 100 times 2 to the power 24 bacteria. We'll not calculate the number over here. That's not important for our present purpose. Our current purpose is to find a formula which can generally tell you what the total number of bacteria or any other cells for that matter are there after a certain amount of time, if we are told how fast they're multiplying. So let's try to find out such a formula. So let's say we want to find out the total number of cells after a given amount of time, right? So we'll call the number of cells in. We'll call the time t. And following that pattern, we can see we always have to multiply in order to get the total number of bacteria. We always have to multiply with the number we started out with. So we'll call that N0. N0 is the original number of cells. And then we will multiply by 2 to the power whatever the time is. So this can be our formula for such a system where every hour the bacteria are forming two cells. Now every type of geometric growth may not be in this way. So every time it might not be doubling. It might be multiplied by some other number. So let's find out what else may happen. Here we have the cross section of a plant. And it has, it's a cross section of a stem and it has primary xylem, secondary xylem, primary phloem and so on. A lot of layers. Here we are interested only in the secondary xylem layer. We want to find out how it grows and we want to quantify it. Let's say the secondary xylem layer originally has a cross sectional area of 0.5 micrometer square. And it grows, that is it's cells divide but we are measuring, in this case we are measuring the growth in terms of the area. So we find out that every one day it grows 1.2 times. That means the area that was there will be multiplied by 1.2. So after 30 days we want to find out what happens after 30 days. How much has it grown? So again let's apply logic. First let's find out what happens after one day. After one day we had 0.5 micrometer square to begin with and we multiply it by 2 because that's number of times it grows. Which is equal to 0.6 micrometer square. We can also write this as 0.5 times 1.2 to the power 1. After 2 days we can take this directly from here and then this multiply with 1.2 again which is 0.7 micrometer square which we can write as 0.5 times 1.2 square. Similarly for 3 days I am not writing all the details but the number is 0.9 micrometer square and how we find it out is we multiply 0.5 by 1.2 to the power 3. So you see it's similar to the example of bacteria that we saw before. So then after 30 days we apply a similar way of calculation. The number will be now 0.5 times 1.2 to the power 30 which comes out to be equal to around 120 micrometer square. So you see the way we are calculating things is very similar to the previous example of the bacteria. Here also we could write a general formula. We could write nt is equal to n0 times. In this case it is 1.2 that is being multiplied again and again. So it is 1.2 which is being raised to the power I'll just write the multiplication sign here and then 1.2 raised to t. Now we want to find a general formula that is applicable to all situations, all systems. And you see that there is a difference in these two here. The base of the power is 2 whereas over here the base of the power is 1.2. So what is the solution for it? So then what we do is in general we tend to write all powers with a base of e. So instead of writing the formulas like this what we write is we say nt is equal to n0 times e to the power rt. So here e to the power r is nothing but 2 in this case and 1.2 in this case. So I'll just write it down here e to the power r is 2 and here e to the power r is 1.2. And e is nothing but a constant which is often used in mathematical calculations. It's an irrational number and its value is around 2.718 and then it goes on. So this is the general formula for geometry growth. nt is equal to n0 times e to the power rt. r is also a constant but it's constant only for the particular system. So naturally for the secondary xylem here the r will be different from the bacteria here. The secondary xylem here will also be different from some other secondary xylem in some other plant because different plants grow at different rates. So if you are given this formula and if you are asked to find out let's say r or t where the values of the other variables are given then you should be able to calculate it very easily. And since there is an exponent involved this type of growth the geometric growth is also called exponential growth. So how would the graph of exponential growth look like? Here I've written down some numbers from the calculations that we just did for the secondary xylem growth. I've taken more numbers that we just calculated because it will make things clearer when you plot the graph. We will plot the area in micrometer square versus the time in days. So when we plot the points that are listed in the table this is what they look like. And let's now join the dots to find out what the graph looks like. It has this particular shape where initially there seems to be a lag. The growth barely seems to be happening here. It's almost a line with zero slope. However as time passes the slope of the line goes on increasing, increasing and over here it's rising really steeply. That's typically how the graph of exponential growth looks like. So we have seen that geometric growth or exponential growth has the formula. Nt is equal to n0 times e to the power rt. N can also be represented in different ways. Some people will write the equation like this. Wt is equal to w0 times e to the rt. So w is nothing but the weight. Sometimes it's written as the height h. It doesn't matter how you write the formula as long as you know what it's representing. It can be the number of cells. It can be the weight of the tissue growing. It can be the height of the plant growing. It can be the area of the tissue growing. And remember r is a constant for whichever plant or whichever organism we're looking at. It's called the growth rate constant. And this is called exponential growth or geometric growth. And the graph looks like this. It has this shape and the slope increases with time.