Difference Between Geometric Sequence and Exponential Function

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.

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Table of Contents

Geometric Sequence vs Exponential Function

The 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

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.

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.

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?

Exponential functions represent dynamic systems, such as the growth of bacteria or decay of matter.

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.

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

A geometric sequence is discrete while an exponential function is continuous.
Geometric sequences can be represented by the general formula a+ar+ar²+ar³where r is the fixed ratio while exponential function has the following formula f(x)= b^xwhere b is base value and x is a real number.

References

https://epubs.siam.org/doi/pdf/10.1137/0130032

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