Math CentralQuandaries & Queries


Question from Moti:

Let {an} be an arithmetic sequence such that its 1st, 20th, and 58th terms are
consecutive terms of some geometric sequence. Find the common ratio of the
geometric sequence.


Hi Moti,

If the first term of the arithmetic sequence is $a_1$ and the common difference id $d$ what is the 20th term? What is the 58th term?

The problem says the $a_1$ is also some term in a geometric sequence, lets call it $b.$ Suppose also that the common ratio of the geometric sequence is $r.$ In a geometric sequence each term after the first is $r$ times the previous term so the next term after $b$ is $b \times r$ and the term after that is $b \times r \times r.$ These three terms are the same as the three terms in the arithmetic sequence mentioned above. This gives toy three equations. Solve them for $r.$

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