Mathematics comes in diverse forms and with different application. Understanding the basics of everything that pertains to a particular topic set the foundation needed to be good at solving any form of question. Exponential growth is one topic that understanding the basics gave me an edge and with that, I was able to solve both linear and compound interest easily.
Having gone through the learning process and understood all the concepts of exponential growth, I strongly believe anyone can too. This has prompt me to share all that I have been able to learn with anyone finding it difficult to understand exponential growth and how it works.
But before I proceed, I will like to go over what exponential growth is really all about.
What is exponential growth and how is it graphically represented?
The first thing I discovered about exponential growth is its simple formula which is y=a \times (b)^x. Understanding this formula set the foundation on which I learnt how to solve other problems related to exponential growth. But before this can be demonstrated graphically, the value of “a” must be positive, while the base exponent which is “b” must be greater than 1.
Unlike other graphical representation, the graph of exponential growth has a upward slope to the right, which signifies that this graph grows by a factor. To make the learning process easier, I will be sharing few examples on how it’s been solved and it’s application to compound interest.
How can i use exponential growth to solve compound interest?
The first thing I always consider before going ahead to solve anything that has to do with exponential growth is the guiding rules already stated above. If both are met, then I can go ahead with solving for compound interest.
To explain this mathematics principle, let’s take the value of a = 2 and b=1.2. Using y=a \times (b)^x formula, I will have y = 2 (1.2)^x. From this, I should be able to calculate the value of y when x=0,1,2,3,4,~. In turn, this will give me the values of y=2, 2.4, and 2.88 respectively.
To help explain the way this was calculated much better, I will be using this same formula to calculate the compound interest of an initial capital of $200 with an interest rate of 8% annually. But before we proceed with that, I will like to show the difference between linear growth and exponential growth by solving for linear growth using $20 annual interest rate.
For Linear Growth:
Unlike exponential growth, the interest growth rate is constant. Which means it grows by a constant value. A good example is when the initial capital of an investment is $200, but as the number of years increases, the total amount of the first year becomes the sum of the initial capital and the interest rate.
As such, the value of the first year y=200 + 20= 220, second year y2=220 + 20= 240. This goes on and on until the specified number of years is reached. Where year 3, 4, 5, 6, 7, and 8 are $260, $280, $300, $320, $340, and $360 respectively.
For Exponential Growth:
Using exponential growth to solve compound interest, all i do is take a 5% annual growth to calculate the total earnings after a period. With the same capital which is $200, the first year will be $210and the subsequent years goes thus, , $220.50, $231.53, $243.11.
This calculation is done using y = 200 (1.05)^x.