Chapter 12

Reciprocals of a Geometric Sequence If \(a_{1}, a_{2}, a_{3}, \ldots\) is a geometric sequence with common ratio \(r,\) show that the sequence $$\frac{1}{a_{1}}, \frac{1}{a_{2}}, \frac{1}{a_{3}}, \ldots$$ is also a geometric sequence, and find the common ratio.

Find the first 40 terms of the sequence defined by $$a_{n+1}=\left\\{\begin{array}{ll}{\frac{a_{n}}{2}} & {\text { if } a_{n} \text { is an even number }} \\ {3 a_{n}+1} & {\text { if } a_{n} \text { is an odd number }}\end{array}\right.$$ and \(a_{1}=11 .\) Do the same if \(a_{1}=25 .\) Make a conjecture about this type of sequence. Try several other values for \(a_{1},\) to test your conjecture.

Logarithms of a Geometric Sequence If \(a_{1}, a_{2}, a_{3}, \ldots\) is a geometric sequence with a common ratio \(r>0\) and \(a_{1}>0,\) show that the sequence $$\log a_{1}, \log a_{2}, \log a_{3}, \ldots$$ is an arithmetic sequence, and find the common difference.

Exponentials of an Arithmetic Sequence If \(a_{1}, a_{2},\) \(a_{3}, \ldots\) is an arithmetic sequence with common difference \(d,\) show that the sequence $$10^{a_{1}}, 10^{a_{2}}, 10^{a_{3}}, \ldots$$ is a geometric sequence, and find the common ratio.