Problem:
The sum of the first 2011 terms of a geometric series is 200. The sum of the first 4022 terms of the same series is 380 . Find the sum of the first 6033 terms of the series.
Solution:
If a is the first term and r is the common ratio of this series, then the nth term of the series is arn−1. It follows that the sum of the terms 2012 through 4022 is r2011 times the sum of the first 2011 terms. Thus 200+200r2011=380, and r2011=109. The sum of the first 6033 terms of the series is 200+200r2011+200(r2011)2=200+200(109)+200(109)2= 200+180+162=542.
The problems on this page are the property of the MAA's American Mathematics Competitions