Formulas like f(x)=x may be used to express functions. The only kind of functions that contains numbers are sequences. Because a geometric function is discrete, and exponential functions are continuous, geometric functions vary from exponential functions. While an exponential function might have different values for the variable x function, a geometric sequence has the same values at all times.
Mathematical growth patterns include exponential function and geometric sequence. Although they seem to be identical at first look, the laws they adhere to are very different.
A common ratio is used to multiply successive integers in a geometric function. In contrast, an exponential function uses a variable exponent to construct a series.
Geometric sequence
To create a geometric sequence, you multiply one integer by the next in the series. The pattern that results from multiplying one number by an integer, say x, and then multiplying the resulting value by another integer, say y, to arrive at a third number is known as a geometric sequence.
A geometric sequence is one in which the ratio of the following integers does not vary during the course of the series. Taking any two consecutive integers from the sequence and dividing one by the other, you’ll always get the same result regardless of how many times you do it.
Identifying the fixed ratio r is necessary in order to obtain the next number in a pattern. Similarly, if a number is absent from the sequence, the fixed ratio may be multiplied by the previous number to determine the omission.
A geometric sequence’s pattern depends on the value of the common ratio r. If r is one, the pattern stays the same; if r is bigger than one, the pattern expands to infinity and back again. A geometric sequence is represented as a discrete graph on the screen.
A geometric sequence may be expressed mathematically as follows:
A+ar+ar2+ar3 and so on are all examples of the sum a+ar. Geometric progression is the process of increasing the number of geometric forms by a predetermined ratio. Only entire numbers may be utilized in a geometric progression.
Exponential function
The mathematical function that may be described by the following formula is an exponential function.
There is an equal and opposite relationship between the two variables. The base number b and an actual number x are used.
In contrast to most functions, the exponent of an exponential function is a variable rather than the base number.
Mathematicians value a particular instance of the exponential function highly. E is the fixed value of the base number in this situation. The number e=2.718 is widely accepted as the best starting point for an exponential series in mathematics.
To put it another way, a function with an exponential exponent has an independent variable x as its exponent to a constant base. Dynamic systems, such as bacterial growth or matter decay, are represented by exponential functions.
A continuous graph may depict an exponential function. It comprises all kinds of numbers, including negative ones. Explosive patterns may be noticed in exponential functions since the value grows rapidly with each succeeding number.
When discussing exponential growth, the exponential function might be utilized. The function’s starting value is multiplied by two for a set amount of time in this case. Because exponential growth is a function that grows exponentially, it may be said to be increasing exceedingly quickly.
A polynomial function will always increase slower than an exponential function, regardless of the situation.
Difference between Geometric Sequence and Exponential Function
- When a base number is multiplied by a variable exponent, it yields an exponential function, which is a function that multiplies succeeding integers with the same common fixed ratio.
- Exponential functions may describe dynamic systems, whereas geometric sequences show the increase in the size of a geometric object.
- While the value of a variable might be negative in a geometric series, it is always a whole integer in an exponential sequence.
- A discrete geometric sequence differs from a continuous exponential function in that the latter is a function of time.
- An exponential function (f(x)=bx) has the formula f(x)=bx (b=the base value) whereas geometric sequences (a+ar+ar+ar2+ar3) may be expressed by formula (r=the fixed ratio).
Conclusion
Mathematicians study sets and sequences a lot. A sequence is formed when only integers are used in a function. There are many distinct sorts of functions. Both the geometric and exponential function sequence systems illustrate exponential growth, which is why they are comparable. Although the formulae for the two systems vary, they are nevertheless distinct.