# Difference between an Arithmetic Sequence and a Geometric Sequence

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, ar ^{2}, ar^{3}, ar^{4} …**

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 n ^{th} term of an arithmetic sequence? **

**Answer:**

a

_{n}= 2n + 1 is the formula for finding the n^{th}term of an arithmetic sequence or the n^{th}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 n ^{th} term of a geometric sequence? **

**Answer:**

a

_{n}= ar^{n − 1 }is the formula for finding the n^{th}term of a geometric sequence where ‘a’ is the first term and ‘d’ is the common ratio of a geometric sequence.

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