Difference between an Arithmetic Sequence and a Geometric Sequence (original) (raw)
Last Updated : 18 Feb, 2024
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:
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 n th 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.