Geometric Sequences
A geometric sequence is recognized because each term is calculated as the previous term time a fixed constant called the common ratio (r).
The n-th term can be calculated as:
[tex]a_n=a_1\cdot r^{n-1}[/tex]Where a1 is the first term and r is the common ratio.
We are not given a1, but we know the value of a2 and r, thus:
[tex]a_2=r\cdot a_1[/tex]Or, equivalently:
[tex]a_1=\frac{a_2}{r}[/tex]Calculating:
[tex]a_1=\frac{\frac{1}{2}}{2}=\frac{1}{4}[/tex]Now calculate a9:
[tex]\begin{gathered} a_9=a_1\cdot r^8 \\ a_9=\frac{1}{4}\cdot2^8 \end{gathered}[/tex]Calculating:
[tex]\begin{gathered} a_9=\frac{1}{4}\cdot256 \\ a_9=64 \end{gathered}[/tex]