Function are formulas that can be expressed in the form of f(x)= x. A sequence is technically a type of function that includes only integers.
Geometric Sequence vs Exponential Function
The main difference between geometric function and exponential function is that a geometric sequence is discrete while an Exponential function is continuous. This means that a geometric sequence has specific values at present at distinct points while an exponential function has varied values for the variable function of x.
Exponential Function and Geometric sequence are both a form of a growth pattern in mathematics. Although they may seem similar at one glance, they are very different in terms of the rules they follow.
Geometric function is achieved by multiplying subsequent numbers by a common ratio. An exponential function on the other hand is a function in which a sequence is formed by a variable exponent.
Comparison Table Between Geometric Sequence and Exponential Function (in Tabular Form)
| Parameter of Comparison | Geometric Sequence | Exponential Function |
|---|---|---|
| Definition | It is a sequence achieved by multiplying subsequent numbers with a common fixed ratio. | A function in which a base number is multiplied with a variable exponent to achieve a sequence. |
| Meaning | A geometric sequence represents the increment in the size of geometric systems, which is why the dimension/ fixed ratio is important. | Exponential function can be seen as a representation of dynamic systems such as the growth of bacteria or decay of matter. |
| Variable | The value of the variable is always a whole number | Value of variable includes real numbers of both negative and positive value. |
| Nature of sequence | The obtained sequence is discrete since values are placed at specific points. | The sequence is continuous as there is an assigned function value for possible values of x. |
| Representation formula | a+ar+ar2+ar3 where r is the fixed ratio | f(x)= bx where b is base value and x is a real number. |
What is Geometric Sequence?
A geometric sequence is a sequence derived by multiplying subsequent figures with a fixed number. In other words, if we begin by taking a certain number and multiply it by a number, say x to get the second number, then multiply the second number by x again, to get the third number, the resultant pattern would be called a geometric sequence.
The characteristic feature of a Geometric sequence is that the ratio of subsequent numbers does not change throughout the sequence. This means, that if you take any two consecutive numbers from the sequence, and divide the greater number by the smaller one or vice versa, the number that would be derived would remain constant for all pairs.
In order to derive the subsequent number of a given pattern, one must identify the fixed ratio r. Similarly, a missing number from the sequence can be derived by multiplying the fixed ratio with the preceding number.
In case of a geometric sequence, the value of common ratio r determines the pattern, for example, if r is one, the pattern remains constant, while if r is greater than one, the pattern shall grow up till infinity. The graph plotted for a geometric sequence is discrete.
Mathematically, a geometric sequence can be represented in the following way;
a+ar+ar2+ar3 and so on. Geometric progression represents the growth of geometric shapes by the fixed ratio, hence the dimension in the sequence matters. Only whole numbers can be used in a geometric progression.
What is Exponential Function?
Broadly, an exponential function is a mathematical function that can be represented by the following formula;
f(x)= bx
where b is the base number and x is a real number.
Unlike most functions, in case of an exponential function, the base number remains constant and the exponent is a variable.
A special case of the exponential function is considered to be quite important in mathematics. In this case, the base number has a fixed value also called e. In calculus, the value e=2.718 is considered to be the most suitable choice for the base number of an exponential sequence.
Hence it can be said, that an exponential function is a function with an independent variable x, as the exponent to a fixed base. Exponential functions represent dynamic systems, such as the growth of bacteria or decay of matter.
Exponential function can be represented by a continuous graph. It includes real numbers including negative values. The pattern seen in exponential functions is also known as explosive patterns since the value increases significantly with each subsequent number.
The exponential function can be used to express the phenomenon of exponential growth. This is characterised by a fixed time period in which the initial value of the function is doubled. Since exponential growth itself is an exponential function, it can be characterised as extremely fast-growing.
It is worth noting that under all circumstances an exponential function will have a better growth rate that a polynomial function.
Main Differences Between Geometric Sequence and Exponential Function
- It is a Geometric sequence achieved by multiplying subsequent numbers with a common fixed ratio whereas an exponential function is a function in which a base number is multiplied with a variable exponent to achieve a sequence.
- A geometric sequence is the representation of the increment in the size of a geometric shape, while exponential function can be a representation of dynamic systems.
- The value of the variable in a geometric sequence is always a whole number, while in case of an exponential sequence it is a real number, including negative values.
- A geometric sequence is discrete while an exponential function is continuous.
- Geometric sequences can be represented by the general formula a+ar+ar2+ar3 where r is the fixed ratio while exponential function has the following formula f(x)= bx where b is base value and x is a real number.
Conclusion
Set and sequences are important topics in mathematics. There are different types of functions however when a function is made up of only integers it forms a sequence. Geometric Sequence and Exponential Functions are two sequence systems that are similar as both represent rapid growth. However, the two systems are represented by different formulas, hence are absolutely distinct.
References
- https://epubs.siam.org/doi/pdf/10.1137/0130032
- https://core.ac.uk/download/pdf/22872312.pdf