If \(\{a_{n}\}\) and \(\{b_{n}\}\) are geometric sequences with common ratios of \(r_1\) and \(r_2\), is \(\{a_{n}b_{n}\}\) a geometric sequence too? If yes, what is the common ratio?
Answer (2)
a_{n+1}b_{n+1} / a_{n}b_{n} =( a_{n+1} / a_{n}) * ( b_{n+1} / b_{n} ) = ( r1 ) * ( r2) => {a_{n}b_{n}} a geometric sequence; the common ratio is ( r1 ) * ( r2) .
The product of two geometric sequences { a n } and { b n } is also a geometric sequence. The common ratio of the new sequence { a n b n } is the product of their common ratios, r 1 r 2 . Thus, { a n b n } has a common ratio of r 1 r 2 .
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