Answer:
[tex]a_{n+1}=0.2a_n[/tex] for all n>0, [tex]a_1=16[/tex]
Step-by-step explanation:
Let [tex]\{a_n\}=\{16,3.2,0.64,0.128,\cdots \}[/tex] be the sequence described.
A geometric sequence has the following property: there exists some r (the ratio of the sequence) such that [tex]\frac{a_{n+1}}{a_n}=r[/tex] forr all n>0.
To find r, note that
[tex]\frac{3.2}{16}=\frac{32}{10(16)}=\frac{2}{10}=\frac{1}{5}=0.2[/tex]
Similarly
[tex]\frac{0.64}{3.2}=\frac{64}{10(32)}=\frac{1}{5}=0.2[/tex]
[tex]\frac{0.128}{0.64}=\frac{1}{5}=0.2[/tex]
Thus [tex]a_{n+1}=r a_n=\frac{a_n}{5}=0.2a_n[/tex] for all n>0, and [tex]a_1=16[/tex]
