# Arithmetic vs Geometric Sequence: The Key Difference You Should Know

Are you confused between arithmetic and geometric sequences? Well, it is normal. You are not the only one who is confused regarding arithmetic vs geometric sequences. Several students get confused between these two sequences. And they search for the difference between both sequences.

Well, the sequence is a systematic grouping of terms, numbers, and events. They are arranged in a specified order. Also, the two kinds of sequences are arithmetic and geometric. They follow a pattern and explain how things follow one another. But, there is a significant difference between both the sequences.

However, do you want to know the difference between them? Do you want to learn the similarities between them as well? So, keep scrolling down this blog. You will get to know the essential information regarding both and **algebra homework help**. Therefore, let’s get started.

**Arithmetic vs Geometric Sequence**

The key difference between both is that the numbers in an arithmetic sequence are set up in a way that the difference between the two consecutive terms stays constant. On the other hand, in a geometric sequence, there is a fixed ratio between the successive terms.

Let’s discuss both sequences in detail. Are you ready?

**Arithmetic Sequence**

It is a series of numbers. In this sequence, there is always the same difference between any two successive terms. In simple words, the previous number in the series is multiplied by a fixed integer to get the next number in the series.

So, the formula of an arithmetic sequence is;

“a, a+d, a+2d, a+3d, a+4d,… so on.”

Therefore, here is:

- a: The first term
- d: Common difference between terms.

To understand it better, have a look at the following example.

For example, 2, 4, 6, 8, 10, 12,… and 40.

However, you can see that each number is greater than the one before. And the difference between one and other numbers is “2”. Therefore, we can now say that the numbers are in an arithmetic progression.

**Geometric Sequence**

Whereas, it is also a series of numbers. But in a geometric sequence, the ratio of any two successive terms is usually the same. In simple words, it means that the next number will be multiplied by the previous one in the series.

Thus, the formula of a geometric sequence is;

“a, ar, ar^2, ar^3,… so on”

- a: The first term
- r: The common ratio between the terms.

So, let’s understand this with the assistance of an instance.

For example, 2, 4, 8, 16, 32, 64, 128,… so on.

However, here in this sequence, if you divide any term, you will get “2”. So, the common ratio between the above sequence is “2”. Therefore, it is a geometric sequence.

Now, let’s discuss the similarities between both of them.

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**Arithmetic vs Geometric Sequence: The Similarities**

Well, there is no doubt that both sequences are different from each other. But, the geometric and arithmetic sequences have a special kind of similarity. Their primary qualities are generally similar. Both sequences are bound to integers or numbers according to a preset rule. In simple words, both are the sequence of numbers that cannot be broken with maintaining the series.

Moving further, let’s discuss the main difference between both.

**Arithmetic vs Geometric Sequence**

Now, you are aware of the basics of arithmetic and geometric sequence. Now, let’s move toward the distinction between both sequences.

*Working Principle*

*Working Principle*

The arithmetic sequence usually represents a collection of numbers. In this sequence, every particular number can be calculated in two ways. Either removing a fixed term from the next number or adding a fixed term to the one before it.

Whereas, the geometric sequence is completely different from the arithmetic sequence. It only deals with division and multiplication. Also, it avoids addition and subtraction. However, the series of numbers is calculated by simply multiplying the previous number by a fixed non-zero value.

*Mathematical Representation*

*Mathematical Representation*

An arithmetic sequence often takes the form of a, a+d, a+2d, a+3d, and so on. While the geometric sequence goes as follows: a, ar, ar^2, ar^3, etc.

*Constant Terminology*

*Constant Terminology*

The difference between two successive terms decides the outcome of an arithmetic series. This difference is symbolized by the letter “d.”

But, the geometric sequence is based totally on a fixed ratio. This ratio is represented by “r”.

*Nature*

*Nature*

There is a linear variation when we discuss an arithmetic series. But, a geometric series of numbers shows an exponential kind of variation at the same time.

*Mode Of Direction*

*Mode Of Direction*

The set of numbers in an arithmetic sequence can grow in either the positive or negative directions. Moreover, the nature and importance of the common difference decide how the entire case will carry out.

Whereas, in the case of a geometric sequence, the direction of growth is not bound to any strict rules. Any series in the geometric progression can equally take both positive and negative directions.

**Arithmetic vs Geometric Sequence: How To Find “nth” Term?**

In the arithmetic sequence, you can find the nth term with;

a + (n-1)d

Here,

- a: The first term
- n: Number of terms.
- d: The common difference.

On the other hand, in the geometric sequence,

a(n) = ar (n-1)

Here,

- a: The first term in the sequence.
- n: Number of terms.
- r: The common ratio.

**Final Words**

To sum up, we have discussed arithmetic vs geometric sequences. We have gone through the main differences between them. However, we can say that both are different from each other. Moreover, arithmetic sequences also help in calculating savings, costs, final increments, etc. Whereas, the geometric sequence measures things like interest and population increase.