The sum of the terms of a finite geometric progression is given by the expression:
, where q (ratio) is different from 1. Some cases in which the ratio q belongs to the interval –1 < q < 1, we verify that when the number of elements n approaches infinity (+∞), the expression whatno tends to zero value. Therefore, replacing whatno by zero in the expression of the sum of terms of a finite PG we will have an expression capable of determining the sum of terms of an infinite PG within the interval –1 < q < 1, note:
Example 1
Determine the sum of the elements of the following PG: 
.
Example 2
The mathematical expression of the sum of terms of an infinite PG is recommended in obtaining the generating fraction of a simple or compound periodic decimal. Watch the demo.
Considering the simple periodic decimal 0.222222..., let's determine its generating fraction.
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Example 3
Let's determine the fraction that gives rise to the following decimal number 0.231313..., classified as a composite periodic decimal.
Example 4
Find the sum of the elements of the geometric progression given by (0.3; 0,03; 0,003; 0,0003; ...).
by Mark Noah
Graduated in Mathematics
Brazil School Team
Progressions - Math - Brazil School
Would you like to reference this text in a school or academic work? Look:
SILVA, Marcos Noé Pedro da. "Sum of the Terms of an Infinite PG"; Brazil School. Available in: https://brasilescola.uol.com.br/matematica/soma-dos-termos-uma-pg-infinita.htm. Accessed on June 28, 2021.
