# difference between arithmetic and geometric sequence

### James

Guys, does anyone know the answer?

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## Difference Between Arithmetic and Geometric Sequence (with Comparison Chart)

The primary difference between arithmetic and geometric sequence is that a sequence can be arithmetic, when there is a common difference between successive terms, indicated by 'd',. On the contrary, when there is a common ratio between successive terms, represented by 'r, the sequence is said to be geometric.

## Difference Between Arithmetic and Geometric Sequence

Last updated on October 21, 2017 by Surbhi S

The sequence is described as a systematic collection of numbers or events called as terms, which are arranged in a definite order. Arithmetic and Geometric sequences are the two types of sequences that follow a pattern, describing how things follow each other. When there is a constant difference between consecutive terms, the sequence is said to be an** arithmetic sequence**,

On the other hand, if the consecutive terms are in a constant ratio, the sequence is **geometric**. In an arithmetic sequence, the terms can be obtained by adding or subtracting a constant to the preceding term, wherein in case of geometric progression each term is obtained by multiplying or dividing a constant to the preceding term.

Here, in this article we are going to discuss the significant differences between arithmetic and geometric sequence.

## Content: Arithmetic Sequence Vs Geometric Sequence

Comparison Chart Definition Key Differences Conclusion

### Comparison Chart

BASIS FOR COMPARISON ARITHMETIC SEQUENCE GEOMETRIC SEQUENCE

Meaning Arithmetic Sequence is described as a list of numbers, in which each new term differs from a preceding term by a constant quantity. Geometric Sequence is a set of numbers wherein each element after the first is obtained by multiplying the preceding number by a constant factor.

Identification Common Difference between successive terms. Common Ratio between successive terms.

Advanced by Addition or Subtraction Multiplication or Division

Variation of terms Linear Exponential

Infinite sequences Divergent Divergent or Convergent

### Definition of Arithmetic Sequence

Arithmetic Sequence refers to a list of numbers, in which the difference between successive terms is constant. To put simply, in an arithmetic progression, we add or subtract a fixed, non-zero number, each time infinitely. If **a** is the first member of the sequence, then it can be written as:

**a, a+d, a+2d, a+3d, a+4d..**

where, a = the first term

d = common difference between terms

**Example**: 1, 3, 5, 7, 9…

5, 8, 11, 14, 17…

### Definition of Geometric Sequence

In mathematics, the geometric sequence is a collection of numbers in which each term of the progression is a constant multiple of the previous term. In finer terms, the sequence in which we multiply or divide a fixed, non-zero number, each time infinitely, then the progression is said to be geometric. Further, if **a** is the first element of the sequence, then it can be expressed as:

**a, ar, ar2, ar3, ar 4 …**

where, a = first term

d = common difference between terms

**Example**: 3, 9, 27, 81…

4, 16, 64, 256..

## Key Differences Between Arithmetic and Geometric Sequence

The following points are noteworthy so far as the difference between arithmetic and geometric sequence is concerned:

As a list of numbers, in which each new term differs from a preceding term by a constant quantity, is Arithmetic Sequence. A set of numbers wherein each element after the first is obtained by multiplying the preceding number by a constant factor, is known as Geometric Sequence.

A sequence can be arithmetic, when there is a common difference between successive terms, indicated as ‘d’. On the contrary, when there is a common ratio between successive terms, represented by ‘r’, the sequence is said to be geometric.

In an arithmetic sequence, the new term is obtained by adding or subtracting a fixed value to/from the preceding term. As opposed to, geometric sequence, wherein the new term is found by multiplying or dividing a fixed value from the previous term.

In an arithmetic sequence, the variation in the members of the sequence is linear. As against this, the variation in the elements of the sequence is exponential.

The infinite arithmetic sequences, diverge while the infinite geometric sequences converge or diverge, as the case may be.

### Conclusion

Hence, with the above discussion, it would be clear that there is a huge difference between the two types of sequences. Further, an arithmetic sequence can be used find out savings, cost, final increment, etc. On the other hand, the practical application of geometric sequence is to find out population growth, interest, etc.

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## 6.2: Arithmetic and Geometric Sequences

## 6.2: Arithmetic and Geometric Sequences

Last updated Oct 6, 2021 6.1: Sequences 6.3: Series

Richard W. Beveridge

Clatsop Community College

Two common types of mathematical sequences are arithmetic sequences and geometric sequences. An arithmetic sequence has a constant difference between each consecutive pair of terms. This is similar to the linear functions that have the form y=mx+b. A geometric sequence has a constant ratio between each pair of consecutive terms. This would create the effect of a constant multiplier.

**Examples**

**Arithmetic Sequence:**

{5,11,17,23,29,35,…}

Notice here the constant difference is 6. If we wanted to write a general term for this sequence, there are several approaches. One approach is to take the constant difference as the coefficient for the n term: an=6n+? Then we just need to fill in the question mark with a value that matches the sequence. We could say for the sequence:

{5,11,17,23,29,35,…}

an=6n−1

There is also a formula which you can memorize that says that any arithmetic sequence with a constant difference d is expressed as:

an=a1+(n−1)d

Notice that if we plug in the values from our example, we get the same answer as before:

an=a1+(n−1)d a1=5,d=6

So, a1+(n−1)d=5+(n−1)∗6=5+6n−6=6n−1

or an=6n−1

If the terms of an arithmetic sequence are getting smaller, then the constant difference is a negative number.

{24,19,14,9,4,−1,−6,…}

an=−5n+29

**Geometric Sequence**

In a geometric sequence there is always a constant multiplier. If the multiplier is greater than 1, then the terms will get larger. If the multiplier is less than 1, then the terms will get smaller.

{2,6,18,54,162,…}

Notice in this sequence that there is a constant multiplier of 3. This means that 3 should be raised to the power of n in the general expression for the sequence. The fact that these are not multiples of 3 means that we must have a coefficient before the 3n

{2,6,18,54,162,…} an=2∗3n−1

If the terms are getting smaller, then the multiplier would be in the denominator:

{50,10,2,0.4,0.08,…}

Notice here that each term is begin divided by 5 (or multiplied by

1 5 ).

{50,10,2,0.4,0.08,….}

an= 50 5n−1 or an= 250 5n or an=50∗( 1 5 )n−1 and so on

**Exercises 6.2**

Determine whether each sequence is arithmetic, geometric or neither.

If it is arithmetic, determine the constant difference.

If it is geometric determine the constant ratio.

1) {18,22,26,30,34,…}

2) {9,19,199,1999,…}

3) {8,12,18,27,…}

4) {15,7,−1,−9,−17,…}

5) { 1 2 , 2 3 , 3 4 , 4 5 , 5 6 ,…}

6) {100,−50,25,−12.5,…}

7) {−8,12,32,52,…} 8) {1,4,9,16,25,…}

9) {11,101,1001,10001,…}

10) {12,15,18,21,24,…}

11) {80,20,5,1.25,…}

12) {5,15,45,135,405,…}

13) {1,3,6,10,15,…} 14) {2,4,6,8,10,…}

15) {−1,−2,−4,−8,−16,…}

16) {1,1,2,3,5,8,13,21,…}

## Difference between an Arithmetic Sequence and a Geometric Sequence

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## Difference between an Arithmetic Sequence and a Geometric Sequence

Last Updated : 06 Oct, 2021

Arithmetic is a mathematical operation that deals with numerical systems and related operations. It’s used to get a single, definite value. The word “Arithmetic” comes from the Greek word “arithmos,” which meaning “numbers.” It is a field of mathematics that focuses on the study of numbers and the properties of common operations such as addition, subtraction, multiplication, and division.

A sequence is a collection of items in a specific order (typically numbers). Arithmetic and geometric sequences are the two most popular types of mathematical sequences. Each consecutive pair of terms in an arithmetic sequence has a constant difference. A geometric sequence, on the other hand, has a fixed ratio between each pair of consecutive terms.

### Arithmetic Sequence

If the difference between any two consecutive terms is always the same, a sequence of integers is termed an Arithmetic Sequence. Simply put, it indicates that the next number in the series is calculated by multiplying the preceding number by a set integer. Further, an Arithmetic Sequence can be written as,

**a, a + d, a + 2d, a + 3d, a + 4d**

where a = the first term

d = common difference between terms.

For example, in the following sequence: 5, 11, 17, 23, 29, 35, …, the constant difference is 6.

### Geometric Sequence

If the ratio of any two consecutive terms is always the same, a sequence of numbers is called a Geometric Sequence. Simply put, it means that the next number in the series is calculated by multiplying a set number by the preceding number. Further, a Geometric Sequence can be expressed as:

**a, ar, ar2, ar3, ar4 …**

where a = first term

d = common difference between terms.

For instance, 2, 6, 18, 54, 162,… The constant multiplier is 3 in this case.

### How can you tell the difference between an Arithmetic sequence and a Geometric sequence?

To tell the difference between arithmetic and geometric sequence, the following points are important,

An arithmetic Sequence is a set of numbers in which each new phrase differs from the previous term by a fixed amount. Geometric Sequence is a series of integers in which each element after the first is obtained by multiplying the preceding number by a constant factor.

When there is a common difference between subsequent terms, represented as ‘d,’ a series can be arithmetic. The sequence is said to be geometric when there is a common ratio between succeeding terms, indicated by ‘r.’

The new term in an arithmetic sequence is obtained by adding or subtracting a fixed value from the previous term. In contrast to geometric sequence, the new term is found by multiplying or dividing a fixed value from the previous term.

The variation between the members of an arithmetic sequence is linear. In contrast, the variation in the sequence’s elements is exponential.

Infinite arithmetic sequences diverge, while infinite geometric sequences converge or diverge, depending on the situation.

**Difference between an arithmetic sequence and a geometric sequence**

S.No. Arithmetic sequence Geometric sequence

1 Arithmetic Sequence is a set of numbers in which each new phrase differs from the previous term by a fixed amount. A geometric sequence is a collection of integers in which each subsequent element is created by multiplying the previous number by a constant factor.

2 Between successive words, there is a common difference. Between successive words, they have the same common ratio.

3 Subtraction or addition are used to get terms. Division or Multiplication are used to get terms.

4 Example: 5, 11, 17, 23, 29, 35,… Example: 2, 6, 18, 54, 162,…

### Sample Problems

**Question 1: What is a Geometric Sequence, and why is it called that?**

**Answer:**

Because the numbers go from one to another by diving or multiplying by a similar value, it’s called a geometric sequence.

**Question 2: Is it possible for an Arithmetic Sequence to also be Geometric?**

**Answer:**

In mathematics, an arithmetic sequence is defined as a sequence in which the common difference, or variance between subsequent numbers, remains constant. The geometric sequence, on the other hand, is characterized by a stable common ratio between subsequent values. As a result, a sequence cannot be both geometric and arithmetic at the same time.

**Question 3: In an arithmetic sequence, what is ‘a’?**

**Answer:**

An arithmetic sequence is a set of terms in which the difference between two succeeding members of the series is a constant term, ‘a’ is the first term of an in the arithmetic sequence.

**Question 4: What is the procedure for determining the nth term of an arithmetic sequence?**

**Answer:**

an = 2n + 1 is the formula for finding the nth term of an arithmetic sequence or the nth term could be written as a + (n – 1) d.

Where ‘a’ is the first term and ‘d’ is common difference of an arithmetic sequence.

**Question 5: What is the procedure for determining the nth term of a geometric sequence?**

**Answer:**

an = arn − 1 is the formula for finding the nth term of a geometric sequence where ‘a’ is the first term and ‘d’ is the common ratio of a geometric sequence.

Guys, does anyone know the answer?