Do you seek the difference between an arithmetic and a geometric sequence? Well, then you just made it to the right place. In this article, we'll be discussing each and every factor that sets both of them apart. However, if you're here for a short session, the quick answer to the query is here.

Difference Between Arithmetic and Geometric Sequence: A Quick Look

The prime difference between an Arithmetic and a Geometric Sequence is that in an arithmetic sequence, the numbers are set in a manner where the difference between the two consecutive terms remains fixed. In the geometric sequence, there is a fixed ratio between any of the successive terms.

If you’re ready for a detailed overview, let us walk through it.

What is an Arithmetic Sequence?

The term Sequence refers to a progression that any set of items follow. In mathematics, when a series of integers follow a pattern where the differences between the consecutive numbers remain constant, they are said to be in Arithmetic Sequence. 

To understand the scenario better, let us walk through an example: consider the first ten natural even numbers. These are 2, 4, 6, 8, 10, 12 ,14 ,16, 18, and 20. If you observe them carefully, you'll notice that each number exceeds the previous one by "2". In other words, the difference between the consecutive numbers is "2" every time. Hence the numbers will be considered to be in an Arithmetic Progression

graph representation arithmetic sequence
Graph representation : arithmetic sequence

The Formula for the Arithmetic Sequence

In order to generalise a series of numbers or integers in arithmetic progression, Johann Carl Friedrich Gauss came up with a formula that looked like "a, a+d, a+2d, a+3d, and so on."

Here, "a" is known as the first term, and "d" denotes the common difference. Any set of numbers which will follow the above-mentioned pattern will eventually fall under the arithmetic sequence category.

What is a Geometric Sequence?

Just like Arithmetic, set of numbers in Geometric Sequence are also bound by a set of rules. However, there are no common differences here; instead, it is all about common ratios. The numbers in Geometric Progression or Sequence follow a typical pattern where the first term is calculated by multiplying the previous one by a certain number which is widely known as a common ratio. 

Note: It is essential to keep in mind that the entire Geometric Sequence bears a collection of non-zero numbers, which means the availability of "0" is not possible.

Let us consider the following series 2, 4, 8, 16, 32, 64, and so on. Here if you divide any term by the one that holds the adjacent previous position, you'll get 2 every time. Hence, the mentioned series is in Geometric Sequence with 2 as the common ratio.

Graph representation: geometric sequence
Graph representation: geometric sequence

The Formula for the Geometric Sequence

A geometric sequence or progression also has its own formula. The general form of representing a geometric series is "a, ar, ar², ar³, and so on". Similar to the arithmetic sequence, "a" here is the first term, while "r" represents the common ratio.

The Differences Between an Arithmetic and a Geometric Sequence

Now that you're well aware of the basic information let us walk through and learn about the difference between arithmetic and geometric sequence.

Working Principle

The arithmetic sequence generally represents a set of numbers in which any specific number can easily be calculated by either subtracting a fixed term from the succeding number or adding a fixed term to the preceding one. 

On the other hand, geometric sequence bears a different rule altogether. The geometric sequence is free from addition and subtraction and deals entirely with division and multiplication. In this type of progression, any number from the series is calculated by simply multiplying the previous number by a constant non-zero number.

Mathematical Representation

The general form of Arithmetic Sequence is a, a+d, a+2d, a+3d, and so on. While the same for the geometric sequence is a, ar, ar², ar³, so on.

Constant Terminology

The deciding factor in an arithmetic sequence is the difference between two consecutive terms. In general, it is denoted by the letter "d" and is known as the common difference.

The geometric sequence, on the contrary, revolves entirely around a fixed ratio. Generally called the common ratio, the term is represented by "r".

Nature

When we talk about an arithmetic sequence, the variation is in linear form. At the same time, the set of numbers in a geometric sequence bear an exponential form of variation.

Mode of Direction

In an arithmetic sequence, the set of numbers can either move towards the positive direction or the negative direction in the number scale. The entire proceeding depends upon the nature and value of the common difference. 

Coming to the geometric sequence, there exists no hard and fast rule about the direction of progress. Any series in the geometric progression is capable of alternatively taking up positive and negative routes simultaneously.

Comparison Chart: Arithmetic Vs Geometric Sequence

ParameterArithmetic SequenceGeometric Sequence
DefinitionIt is defined as a series where consecutive  numbers follow a fixed difference moving forward.It is a sequence of numbers where each new term can be obtained by multiplying it with a fixed non-zero value throughout.
General Forma,a+d, a+2d, a+3d, and so ona, ar, ar², ar³, so on
Prime FactorConstant Difference (d)Constant Ratio (r)
NatureLinearExponential
Progress DirectionEither Positive or NegativeNo Fixed Rule

Similarities: Arithmetic Vs Geometric Sequence

Although different in every possible aspect, the arithmetic and geometric sequences hold a unique kind of similarity. The similarity basically lies in their prime nature. Both the sequences bound numbers or integers in a certain rule. In other words, they both correspond to a path of a specific chain that can't be broken while keeping the series intact. 

Finding the nᵗʰ Term: Arithmetic Vs Geometric Sequence

For the Arithmetic Sequence,

  • The nᵗʰ term is "a + (n-1)d"
  • a=the first term in the sequence
  • d=the common difference between terms
  • n=number of terms

For the Geometric Sequence,

  • The nᵗʰ term is a(n) = ar(n-1)
  • a=the first term in the sequence
  • r=the common ratio
  • n=number of terms

Frequently Asked Questions

🕵️‍♂️ How can you tell the difference between an arithmetic and a geometric sequence?

The main difference between an arithmetic and a geometric sequence is the rule by which the numbers in the respective series operate. In arithmetic progression, the set of numbers maintain a constant difference, while in the case of geometric progression, the series revolves around a fixed ratio.

📐 Can a sequence be both arithmetic and geometric?

Yes, there exists a possibility when a particular series is arithmetic and geometric at the same time. For example, if we consider a series 2,2,2,2,2 here, the difference is 0 and constant, and similarly, the ratio is fixed at 1. Hence, it is safe to top consider the series to be operating as AP and GP at the same time. 

However, it is essential to keep in mind that the difference between arithmetic and geometric sequence holds tight for this situation as well. It is only the orientation of sequence that resulted in both the sequences embedding in one series.

4️⃣ What are the four kinds of sequences in mathematics?

The four kinds of sequences in mathematics include Arithmetic Sequences, Geometric Sequences, Harmonic Sequences, Fibonacci Numbers.

Conclusion

In this article, we've talked about the difference between arithmetic and geometric sequence in the most comprehensive manner. Quick run through the article, and you'll understand the factors that make these two sequences stand apart. Additionally, we've mentioned a few other essential information that you need to know while sketching arithmetic vs geometric sequence in mind.

In case you face any trouble, make sure to drop a comment, and we'll make sure to come up with a solution real soon.