Introduction
Functions are one way to minimize redundancy and human labor, both in the fields of mathematics and coding. Formulating a concept can greatly help in reducing the extra effort to break a sweat during textbook calculations or real-life problems. Two such concepts are Geometric Sequences and Exponential Functions.
Geometric Sequences vs Exponential Functions
The main difference between a geometric sequence and an exponential function lies in the representation of their function. A geometric function when represented is a discrete function whereas an exponential function when represented is a continuous function. A discrete function has a unique value at separate points whereas a continuous function does show any abrupt changes in function values i.e., discontinuities.
Difference Between Geometric Sequences and Exponential Functions in Tabular Form
| Parameters of Comparison | Geometric Sequences | Exponential Functions |
| Definition | The values of the sequence are typically always positive numbers. | It is a function that contains a base that can be varied through an exponent to generate a sequence. |
| Sign | The values of the sequence are typically always positive numbers. | Exponential functions can result in either a negative number or a positive number. |
| Fundamentals | The sequence is obtained by multiplying the common ratio decided for that sequence, by the present number to obtain the next number. | The function is obtained by varying the exponent with a constant base and is typically represented through a graph. |
| Nature of graphical representation | Geometric sequences behaves as discrete functions when they are represented graphically. | Exponential functions behave as continuous functions when they are represented graphically. |
| Formula | The geometric sequence is defined by the formula f = a + ar + ar^2 + ar^3 + ..., where "a" is the initial term and "r" is the common ratio that is continuously multiplied. | It is a sequence of numbers that maintains a common fixed ratio throughout the sequence. |
| Examples | Geometric sequences can be applied in real-life situations such as Finances, Generation of music notes, Physics problems such as ball-bouncing, etc. | Exponential functions can be used to model and calculate population increase and decrease census, radioactive decay, compound interest, and carbon date artifacts. |
What are Geometric Sequences?
General Form of a Geometric Sequence:-
Let us look at the general form of how a geometric sequence can be represented:
Consider a value ‘a’, which is the base and will remain constant throughout the series.
Another value ‘r’, which is the common ratio and will keep varying to achieve the series.
Thus, the sequence will progress as a, ar, ar2, ar3, ... and so on.
Every series has the last term but it is not easy to predict the last term without knowing the number of terms that exist in the series. Thus, to solve this problem, let us consider a different approach to find the last term of the series.
The nth term of the sequence
The nth term or the last term of a sequence can be calculated using basic principles. Since the first term has no multiplication by the common ratio, we can consider the exponent of the ratio to be zero. The second term is obtained by multiplying the base with the ratio once. Therefore, we can consider the exponent of the ratio for the second term to be 1. Thus,
1st term: a * r^0
2nd term: a * r^1
3rd term: a * r^2, and so on!
Hence, we can determine that the last term, i.e., the nth term, will have an exponent of the ratio of 'n-1'. This means the last term can be calculated as an = a * r^(n-1).
The Common Ratio of the Sequence
The common ratio of a sequence is what forms the subsequent values of the sequence. You have two methods to acquire a sequence through a common ratio.
Either you apply the common ratio starting from the second term and keep multiplying the current number with the common ratio to obtain the next number or
You consider the base as the first term and apply the ratio to the first term to obtain the second term, and apply the ratio squared to the first term to obtain the third term— ar².
Some situations or problems already provide you with the geometric sequence and require you to find the common ratio. How exactly do we calculate this ratio?
We can use the reverse strategy and reach the formula:
Common ratio= (Current term/ preceding term)
Let us consider the current term=an
Preceding term= an-1
Hence, Common ratio= an/an-1= ar^n-1/ar^n-2= r
Sum of ‘n’ Terms in a Geometric Sequence:-
In common problems involving compound interest, it is required to find the sum of a geometric sequence. Let us represent this as,
Sn=a+ar+ar²+ar³+......+ar^n-1
How can Sn be calculated?
The sum of n terms in a geometric sequence is Sn=a(r^n-1)/(r-1) in case r≠1 and r>1.
If r=1, then the sum of terms in a geometric sequence can be calculated as Sn=(n.a).
Properties of a Geometric Sequence
Any three non-zero, consecutive terms in a geometric sequence, say a,b,c, follow the rule b²=a.c
For example: Let us consider the first 3 terms of the general form of a geometric sequence: a,ar,ar².
Then (ar)²=a.ar². Here, both the LHS and RHS turn out to be equal, proving the above-stated property.
A geometric sequence can either be represented as a,ar,ar² or a/r,a,ar.
The product of terms of a geometric sequence that are equidistant from the beginning and the end remains constant. This can be represented as an.a₁=an-1.a₂=an-2.a₃=… and so on.
If each term in the entire geometric sequence is multiplied or divided by a non-zero value, the common ratio remains the same and the resulting sequence will still be a geometric sequence.
If each term in the entire geometric sequence is raised to the power of a non-zero value, the resulting sequence is also geometric.
If each term in a geometric sequence is converted into a logarithmic form, the resulting sequence is an arithmetic sequence, considering that the original geometric sequence only consists of positive terms.
If a₁,a₂,a₃ are terms of a geometric sequence then log(a₁), log(a₂), log(a₃) must be in an arithmetic sequence.
Types of Geometric Sequence
A geometric sequence can be categorized into two types based on the number of terms the sequence consists of: Finite Geometric Sequence and Infinite Geometric Sequence
Finite Geometric Sequence: Deriving its name from its definition, a finite geometric sequence consists of a finite number of terms and is represented as a,ar,ar²,ar³ and the sum of such a series is given by Sn=a+ar+ar²+ar³+......+ar^n-1.
Infinite Geometric Sequence: An infinite geometric sequence does not have a last term and consists of infinite terms. The sequence can be represented as a,ar,ar²,ar³....,ar^n-1.... and the sum of terms can be represented as Σ(t=0 to ∞)(ar^t) = a/(1-r), assuming |r| < 1.
Exponential Growth Functions:-
The quantity involved in an exponential growth function grows very slowly at the beginning stages and then steepens up rapidly. The rate of change increases over time which eventually leads to an increase in the rate of growth. This can be represented as f(x)=ab^x where b>1.
This can be approximated to f(x)=a(1+r)^x
Where:
a = base value of the function
r is the percentage change in decimal form
x is the exponent value of the function
These models of exponential growth can be used to solve problems involving population growth, compound interest or doubling time, etc.
Thus in the context of population growth, we can use the formula P=P0e^kt
Where:
P0 is the initial measure of the population
k is the rate of growth
t is the time duration that is being considered for that particular problem.
Exponential Decay Function
The behavior of an exponential decay function is completely contrary to that of an exponential growth function. The quantity involved in an exponential decay function decreases very rapidly in the initial stages and then decreases very slowly. The rate of change decreases over time and becomes very slow. This can be represented as f(x)=ab^x where 0<b<1.
This can be approximated to f(x)=a(1-r)^x
Where:
a = base of the function
r is the percentage change represented in the decimal form
x is the exponent value of the function.
These models of exponential decay can be used to solve problems that involve population decay, half-life calculations, etc.
Hence when calculating patterns of population decay, we can use the formula P=P0e^-kt
Where:
P0 is the initial measure of the population
k is the rate of decay
t is the time duration that is given in the problem.
Main Differences Between Geometric Sequences and Exponential Functions (In Points)
- A geometric sequence is a sequence of values that maintain a fixed common ratio throughout the series. In contrast, exponential functions are obtained through a constant base and a varying exponential value.
- A geometric sequence usually consists of values that are positive numbers. On the other hand, the values obtained by an exponential function can be both positive values and negative values.
- The geometric function is formed by multiplying the common ratio to the present number to obtain the next number. An exponential is obtained by fixing a base value and graphically changing the exponent to represent it.
- Thus, when talking in terms of graphical representations, geometric sequences behave as discrete functions, whereas exponential functions behave as continuous functions.
- The general form of a geometric sequence is, whereas the general form of an exponential function is
- A few examples of real-life applications of geometric sequences are finances, the generation of music notes, generic physics problems, etc. On the other hand, exponential functions are used to calculate population growth and decay problems, radioactive decay problems, compound interest calculation, etc.
Conclusion
From the above discussion, we can conclude that geometric sequences differ a lot from exponential functions until their very fundamentals. While geometric sequences serve us considerably in real-life problems, exponential functions have a significantly wider range of applications and sub-concepts when compared to their contextual counterparts. Both these concepts form functions and can be correlated through mathematical and graphical representations.
