in proportion when p:q?q:r. This is extended to four terms p, q, r and s as p:q?q:r?r:s, and so on. Sequences that have the property that the ratios of consecutive terms are equal are called geometric progressions. Definitions 9 and 10 apply this, saying that if p, q and r are in proportion then p:r is the duplicate ratio of p:q and if p, q, r and s are in proportion then p:s is the triplicate ratio of p:q.

Number of terms and use of fractions

In general, a comparison of the quantities

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