# Difference Between Arithmetic and Geometric Sequence

You must have been to movie theatres to watch movies with your friends or family. While booking your tickets, have you ever noticed how the seating arrangements are generally made at the movie theatre?

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

This seating arrangement is usually 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 played with some ball 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 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 standard ratio.

## Key Takeaways

1. Arithmetic sequence is a sequence where each term is obtained by adding a constant to the preceding term.
2. Geometric sequence is a sequence where each term is obtained by multiplying a constant by the preceding term.
3. Arithmetic sequence is used to model linear relationships, while the geometric sequence is used to model exponential relationships.

## Arithmetic vs Geometric Sequence

The variation between the members of an arithmetic sequence is linear, while the variation in the geometric sequence’s elements is exponential. Infinite arithmetic sequence diverges, and on the other hand infinite geometric sequences converge or diverge, depending on the situation.

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

## What is Arithmetic Sequence?

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

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

Like anything else in mathematics, an arithmetic sequence also has a formula. The formula for an arithmetic sequence is a+d, a+2d, a+3d, and so on. In this formula, “a” is the first term and “d” is the typical difference between 2 consecutive terms.

You need to know that the behaviour of an arithmetic sequence depends a lot on the common difference. The terms will grow positively if the common difference or the “d” in the formula is positive.

However, the terms will grow negatively if the common difference is negative.

## What is a Geometric Sequence?

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

For example, the sequence 3, 6, 12, 24, and so on is a geometric sequence with the standard ratio being 2.

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

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

If the standard ratio is positive, the terms will also be favourable.

However, if the standard ratio is negative, the terms will alternate between negative and positive. If the standard ratio exceeds 1, the growth will be exponential towards positive or negative infinity.

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

## Main Differences Between Arithmetic and Geometric Sequence

1. An arithmetic sequence is a number sequence 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 the typical difference represented by “d”. The number by which terms multiply or divide in a geometric sequence is the standard 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 exponential.
4. Depending on the common difference, the numbers may progress positively or negatively in an arithmetic sequence. In a geometric sequence, there is no such rule as the numbers may progress positively and negatively in the same sequence.
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### 1 thought on “Difference Between Arithmetic and Geometric Sequence”

1. 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.