1. In a geometric series with first term, a and common ratio, r (where r is real and r#1), the sum of the first 7 terms is 4 times the sum of the following 7 terms. Find the ratio of the sum of the first 21 terms to the sum of the first 14 terms.​

9514 1404 393

  21/20

Step-by-step explanation:

Let Sa, Sb, and Sc represent the sums of the first 7 terms, the next 7 terms, and the 7 terms following that. The problem statement tells us ...

  Sa/Sb = 4/1

We know from the nature of geometric series that this means ...

  Sb/Sc = 4/1

Then the desired ratio is ...

  S21/S14 = (Sa +Sb +Sc)/(Sa +Sb) = (Sa +Sa/4 +(Sa/4)/4)/(Sa +Sa/4)

  = (1 +1/4 +1/16)/(1 +1/4) = (16 +4 +1)/(16 +4) . . . . factor out Sa; multiply by 16/16

  S21/S14 = 21/20

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