The sequence is described as a systematic collection of numbers or events called as terms, which are arranged in a definite order. Arithmetic and Geometric sequences are the two types of sequences that follow a pattern, describing how things follow each other. When there is a constant difference between consecutive terms, the sequence is said to be an** arithmetic sequence**,

On the other hand, if the consecutive terms are in a constant ratio, the sequence is **geometric**. In an arithmetic sequence, the terms can be obtained by adding or subtracting a constant to the preceding term, wherein in case of geometric progression each term is obtained by multiplying or dividing a constant to the preceding term.

Here, in this article we are going to discuss the significant differences between arithmetic and geometric sequence.

## Content: Arithmetic Sequence Vs Geometric Sequence

### Comparison Chart

Basis for Comparison | Arithmetic Sequence | Geometric Sequence |
---|---|---|

Meaning | Arithmetic Sequence is described as a list of numbers, in which each new term differs from a preceding term by a constant quantity. | Geometric Sequence is a set of numbers wherein each element after the first is obtained by multiplying the preceding number by a constant factor. |

Identification | Common Difference between successive terms. | Common Ratio between successive terms. |

Advanced by | Addition or Subtraction | Multiplication or Division |

Variation of terms | Linear | Exponential |

Infinite sequences | Divergent | Divergent or Convergent |

### Definition of Arithmetic Sequence

Arithmetic Sequence refers to a list of numbers, in which the difference between successive terms is constant. To put simply, in an arithmetic progression, we add or subtract a fixed, non-zero number, each time infinitely. If **a** is the first member of the sequence, then it can be written as:

**a, a+d, a+2d, a+3d, a+4d..**

where, a = the first term

d = common difference between terms

**Example**: 1, 3, 5, 7, 9…

5, 8, 11, 14, 17…

### Definition of Geometric Sequence

In mathematics, the geometric sequence is a collection of numbers in which each term of the progression is a constant multiple of the previous term. In finer terms, the sequence in which we multiply or divide a fixed, non-zero number, each time infinitely, then the progression is said to be geometric. Further, if **a** is the first element of the sequence, then it can be expressed as:

**a, ar, ar ^{2}, ar^{3}, ar^{ 4} …**

where, a = first term

d = common difference between terms

**Example**: 3, 9, 27, 81…

4, 16, 64, 256..

## Key Differences Between Arithmetic and Geometric Sequence

The following points are noteworthy so far as the difference between arithmetic and geometric sequence is concerned:

- As a list of numbers, in which each new term differs from a preceding term by a constant quantity, is Arithmetic Sequence. A set of numbers wherein each element after the first is obtained by multiplying the preceding number by a constant factor, is known as Geometric Sequence.
- A sequence can be arithmetic, when there is a common difference between successive terms, indicated as ‘d’. On the contrary, when there is a common ratio between successive terms, represented by ‘r’, the sequence is said to be geometric.
- In an arithmetic sequence, the new term is obtained by adding or subtracting a fixed value to/from the preceding term. As opposed to, geometric sequence, wherein the new term is found by multiplying or dividing a fixed value from the previous term.
- In an arithmetic sequence, the variation in the members of the sequence is linear. As against this, the variation in the elements of the sequence is exponential.
- The infinite arithmetic sequences, diverge while the infinite geometric sequences converge or diverge, as the case may be.

### Conclusion

Hence, with the above discussion, it would be clear that there is a huge difference between the two types of sequences. Further, an arithmetic sequence can be used find out savings, cost, final increment, etc. On the other hand, the practical application of geometric sequence is to find out population growth, interest, etc.

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