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, 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

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.