Difference Between Arithmetic and Geometric Sequence

All of you must have been to movie theaters to watch movies with your friends or family members. While booking your tickets, have you ever noticed the way the seating arrangements are normally made at the movie theater?

The number of seats in the previous row will always be lesser than the next row by a specific number.

This seating arrangement is normally in an arithmetic sequence. Thus, it can be said that a sequence that decreases or increases by a constant number is known as an arithmetic sequence.

On the other hand, a geometric sequence is something completely different. Most of you have played with some sort of balls during your childhood days.

Whether you use a football or a basketball, you will notice that the height at which it bounces tends to decrease every time it hits the ground. This decrease in the bouncing height is in a geometric sequence.

Thus, it can be said that the geometric sequence is basically a sequence in which each term multiplies or divides by the same value from one specific term to the next one. The value by which a term divides or multiplies is known as the common ratio.

Arithmetic vs Geometric Sequence

The difference between Arithmetic and Geometric Sequence is that while an arithmetic sequence has the difference between its two consecutive terms remains constant, a geometric sequence has the ratio between its two consecutive terms remains constant.

Arithmetic vs Geometric Sequence

The difference between two consecutive terms in an arithmetic sequence is referred to as the common difference. On the other hand, the ratio of two consecutive terms in a geometric sequence is referred to as the common ratio.

Comparison Table

Parameter of ComparisonArithmetic SequenceGeometric Sequence
DefinitionIt is a list of numbers, in which every new term alters from another preceding term by a definite quantity.It is a sequence of numbers in which each new term is calculated by multiplying by a non-zero and fixed number.
Calculated ByAddition or SubtractionMultiplication or Division
Identified ByA constant difference between 2 successive terms.A common ratio between 2 successive terms.
FormLinear FormExponential Form

What is Arithmetic Sequence?

When you talk about arithmetic sequence or arithmetic progression, it basically refers to a sequence of different numbers in which the difference between 2 consecutive numbers is always constant.

In this type of sequence, difference means the first term subtracted from the second term. If you consider a sequence such as 1, 4, 7, 10, 13…it is an arithmetic sequence in which the constant difference if 3.

Just like anything else in mathematics, an arithmetic sequence also has a formula. The formula used to find an arithmetic sequence is a, a+d, a+2d, a+3d, and so on. In this formula, “a” is the first term and “d” is the common difference between 2 consecutive terms.

It is important for you to know that the behavior of an arithmetic sequence depends a lot on the common difference. If the common difference or the “d” in the formula is positive, then the terms will grow in a positive manner.

However, if the common difference is negative, the terms will grow in a negative manner.

arithmetic sequence

What is a Geometric Sequence?

The geometric sequence or geometric progression in mathematics happens to be a sequence of different numbers in which each new term after the previous is calculated by simply multiplying the previous term with a common ratio. This common ratio is a fixed and non-zero number.

As an example, the sequence 3, 6, 12, 24, and so on is a geometric sequence with the common ratio being 2.

A geometric sequence also has a formula of its own. The normal form of a geometric sequence is in the form of a, ar, ar², ar³, ar4 and so on.

When you need to find the n-th term in any geometric sequence, the formula to use is an = arn-1, where the common ratio “r” and the initial value “a” will be given. There are certain factors you should remember when it comes to a geometric sequence.

If the common ratio is positive, the terms will also be positive.

However, if the common ratio is negative, the terms will be alternate between negative and positive. If the common ratio is greater than 1, the growth will be in an exponential form towards positive or even negative infinity.

If the common ratio is 1, then the progression will be a constant sequence.

geometric sequence

Main Differences Between Arithmetic and Geometric Sequence

  1. An arithmetic sequence is a sequence of numbers that is calculated by subtracting or adding a fixed term to/from the previous term. However, a geometric sequence is a sequence of numbers where each new number is calculated by multiplying the previous number by a fixed and non-zero number.
  2. The difference between two consecutive terms in an arithmetic sequence is known as the common difference that is represented by “d”, and the number by which terms multiple or divide in a geometric sequence is known as the common ratio represented by “r”.
  3. When it comes to an arithmetic sequence, the variation is in a linear form. On the other hand, when it comes to a geometric sequence, the variation is in an exponential form.
  4. In an arithmetic sequence, the numbers may either progress in a positive or negative manner depending upon the common difference. Whereas, in a geometric sequence there is no such rule as the numbers may progress alternatively in a positive and negative manner in the same sequence.
Difference Between Arithmetic and Geometric Sequence

References

  1. https://arxiv.org/pdf/1001.5055
  2. https://msp.org/pjm/1971/38-2/pjm-v38-n2-p05-s.pdf
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Nikhil
Nikhil

I am a Math teacher and I love to solve equations. I came here searching for Arithmetic vs Geometric Sequence. The common difference and common ratio have been written out so that anyone can understand. Good work.