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The first term of an infinite geometric progression is 5 and the sum of its terms is 20. What is the common ratio of the progression?

User Kutomer
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Answer:

  • The common ratio of the progression is 3/4

Step-by-step explanation:

A geometric progression is a sequence of terms in which the consecutive terms have a constant ratio; thus, each term is equal to the previous one multiplied by a constant value:


First\ term=a_1\\\\ Second\ term=a_2=a_1* r\\\\ Third\ term=a_3=a_2* r=a_1* r^2\\\\n_(th)\ term=a_n=a_(n-1)* r=a_1* r^(n-1)

A infinite geometric progression may have a finite sum. When the absolute value of the ratio is less than 1, the sum of the infinite geometric progression has a finite value equal to:


  • S_(\infty)=(a_1)/(1-r)

Thus, the information given translates to:


a_1=5\\ \\ S_(\infty)=20=(5)/(1-r)

Now you can solve for the constant ratio, r:


1-r=(5)/(20)\\ \\ r=1-(5)/(20)\\ \\ r=(15)/(20)\\  \\ r=3/4

User Vaibhav Deshmukh
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