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Guilherme Mazui

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What is the Difference Between Arithmetic and Geometric Series?

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The main difference between arithmetic and geometric sequences lies in the relationship between their consecutive terms.

Arithmetic Sequence:

  • In an arithmetic sequence, there is a constant difference between each consecutive pair of terms.
  • The difference between any two consecutive terms is always the same, and it can be positive, negative, or zero.
  • The general formula for an arithmetic sequence is: $$a, a + d, a + 2d, a + 3d, a + 4d, \dots$$, where $$a$$ is the first term and $$d$$ is the common difference between terms.
  • Examples of arithmetic sequences include: $$2,4,6,8,10,12,14,16,18,20$$ or $$5,11,17,23,29,35,41,47,53,67,73$$.

Geometric Sequence:

  • In a geometric sequence, there is a constant ratio between each consecutive pair of terms.
  • The ratio between any two consecutive terms is always the same, and it can be greater than 1, less than 1, or equal to 1.
  • The general formula for a geometric sequence is: $$a, ar, ar^2, ar^3, ar^4, \dots$$, where $$a$$ is the first term and $$r$$ is the common ratio.
  • Examples of geometric sequences include: $$2,4,8,16,32,64,128,256,512,1024$$ or $$5,15,45,135,405,1215$$.

In summary, arithmetic sequences have a constant difference between consecutive terms, while geometric sequences have a constant ratio between consecutive terms. The variation in the elements of an arithmetic sequence is linear, while the variation in the elements of a geometric sequence is exponential.

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  • What is the Difference Between Arithmetic and Geometric Series?
  • Comparative Table: Arithmetic vs Geometric Series

    • Comparative Table: Arithmetic vs Geometric Series

    Here is a table highlighting the differences between arithmetic and geometric series:

    Feature Arithmetic Series Geometric Series
    Definition A sequence of integers in which the difference between consecutive terms is constant. A sequence of numbers in which the ratio of consecutive terms is constant.
    Common Difference/Ratio Each consecutive pair of terms has a constant difference. Each consecutive pair of terms has a constant ratio.
    General Term $$a_n = a + (n - 1)d$$, where $$a$$ is the first term and $$d$$ is the common difference. $$a_n = a \cdot r^{n-1}$$, where $$a$$ is the first term and $$r$$ is the common ratio.
    Examples 5, 11, 17, 23, 29, 35, … (with a common difference of 6). 2, 6, 18, 54, … (with a common ratio of 3).

    In an arithmetic series, the difference between consecutive terms is always the same, while in a geometric series, the ratio between consecutive terms is always the same.

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    Guilherme Mazui

    Guilherme Mazui is graduated in journalism from the Federal University of Minas Gerais (UFMG) and a master's degree in Communication from the University of São Paulo (USP). In addition, he has experience in advertising writing and has worked as a content editor in several companies.