## What is the Difference between Arithmetic and Geometric Sequence

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Both arithmetic and geometric sequences follow a **pattern**, so they are similar. The next number is found in an **arithmetic** sequence by **adding** or **subtracting** the same number. In the same way, the following number in a **geometric** sequence is found by **multiplying** or **dividing** the same number. But the two kinds of sequences are very different from each other.

In this article, we will talk about the big **differences** between an arithmetic sequence and a geometric sequence.

### Chart of Differences

Comparative Points | Arithmetic Sequence | Geometric Sequence |

Meaning | An arithmetic sequence is referred to a list of numbers in which each new term differs from the previous term adding by a fixed amount. |
A geometric sequence is a series of numbers where each subsequent number is obtained by multiplying the one before it by a fixed amount. |

Identification | Common Difference | Common Ratio |

Advanced by | Addition or Subtraction | Multiplication or Division |

Variation of terms | Linear | Exponential |

Infinite sequences | Divergent | Divergent or Convergent |

### Exercises for Comparing Arithmetic and Geometric Sequences

**For each sequence, state if it is arithmetic, geometric, or neither:**

**1) **\(-5, \ 15, \ -45, \ 135, \ -405, \ ...\)

**2) **\(7, \ 12, \ 17, \ 22, \ 27, \ ...\)

**3) **\(56, \ 49, \ 42, \ 35, \ 28, \ ...\)

**4) **\(78, \ 90, \ 102, \ 114, \ 126, \ ...\)

**5) **\(48, \ 24, \ 12, \ 6, \ 3, \ ...\)

**6) **\(192, \ 48, \ 12, \ 3, \ \frac{3}{4} \ , \ ...\)

**7) **\(-162, \ 54, \ -18, \ 6, \ -2, \ ...\)

**8) **\(59, \ 46, \ 33, \ 20, \ 7, \ ...\)

**9) **\(23, \ 39, \ 55, \ 71, \ 87, \ ...\)

**10) **\(\frac{1}{18} \ , \ \frac{1}{9} \ , \ \frac{2}{9} \ , \ \frac{4}{9} \ , \ \frac{8}{9} \ , \ ...\)

**For each sequence, state if it is arithmetic, geometric, or neither:**

**1) **\(-5, \ 15, \ -45, \ 135, \ -405, \ ...\)

\(\color{red}{r \ = \ \frac{15}{-5} \ = \ -3 \ = \ \frac{-45}{15} \ ⇒ \ geometric}\)

**2) **\(7, \ 12, \ 17, \ 22, \ 27, \ ...\)

\(\color{red}{d \ = \ 12 \ - \ 7 \ = \ 5 \ = \ 17 \ - \ 12 \ ⇒ \ arithmetic}\)

**3) **\(56, \ 49, \ 42, \ 35, \ 28, \ ...\)

\(\color{red}{d \ = \ 49 \ - \ 56 \ = \ -7 \ = \ 42 \ - \ 49 \ ⇒ \ arithmetic}\)

**4) **\(78, \ 90, \ 102, \ 114, \ 126, \ ...\)

\(\color{red}{d \ = \ 90 \ - \ 78 \ = \ 12 \ = \ 102 \ - \ 90 \ ⇒ \ arithmetic}\)

**5) **\(48, \ 24, \ 12, \ 6, \ 3, \ ...\)

\(\color{red}{r \ = \ \frac{24}{48} \ = \ \frac{1}{2} \ = \ \frac{12}{24} \ ⇒ \ geometric}\)

**6) **\(192, \ 48, \ 12, \ 3, \ \frac{3}{4} \ , \ ...\)

\(\color{red}{r \ = \ \frac{48}{192} \ = \ \frac{1}{4} \ = \ \frac{12}{48} \ ⇒ \ geometric}\)

**7) **\(-162, \ 54, \ -18, \ 6, \ -2, \ ...\)

\(\color{red}{r \ = \ \frac{54}{-162} \ = \ -\frac{1}{3} \ = \ \frac{-18}{54} \ ⇒ \ geometric}\)

**8) **\(59, \ 46, \ 33, \ 20, \ 7, \ ...\)

\(\color{red}{d \ = \ 46 \ - \ 59 \ = \ -13 \ = \ 33 \ - \ 46 \ ⇒ \ arithmetic}\)

**9) **\(23, \ 39, \ 55, \ 71, \ 87, \ ...\)

\(\color{red}{d \ = \ 39 \ - \ 23 \ = \ 16 \ = \ 55 \ - \ 39 \ ⇒ \ arithmetic}\)

**10) **\(\frac{1}{18} \ , \ \frac{1}{9} \ , \ \frac{2}{9} \ , \ \frac{4}{9} \ , \ \frac{8}{9} \ , \ ...\)

\(\color{red}{r \ = \ \frac{\frac{1}{9}}{\frac{1}{18}} \ = \ 2 \ = \ \frac{\frac{2}{9}}{\frac{1}{9}} \ ⇒ \ geometric}\)