Problem:
A sequence of three real numbers forms an arithmetic progression with a first term of . If is added to the second term and is added to the third term, the three resulting numbers form a geometric progression. What is the smallest possible value for the third term of the geometric progression?
Answer Choices:
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Solution:
The terms of the arithmetic progression are , and for some real number . The terms of the geometric progression are , and . Therefore
Thus or . The corresponding geometric progressions are 9,21 , 49 and , so the smallest possible value for the third term of the geometric progression is .
The problems on this page are the property of the MAA's American Mathematics Competitions