Question

Complete the recursive formula of the geometric sequence 16\,,\,3.2\,,\,0.64\,,\,0.128,...16,3.2,0.64,0.128,

Answer 1

Answer:

$a_{n+1}=0.2a_n$ for all n>0, $a_1=16$

Step-by-step explanation:

Let $\{a_n\}=\{16,3.2,0.64,0.128,\cdots \}$ be the sequence described.

A geometric sequence has the following property: there exists some r (the ratio of the sequence) such that $\frac{a_{n+1}}{a_n}=r$ forr all n>0.

To find r, note that

$\frac{3.2}{16}=\frac{32}{10(16)}=\frac{2}{10}=\frac{1}{5}=0.2$

Similarly

$\frac{0.64}{3.2}=\frac{64}{10(32)}=\frac{1}{5}=0.2$

$\frac{0.128}{0.64}=\frac{1}{5}=0.2$

Thus $a_{n+1}=r a_n=\frac{a_n}{5}=0.2a_n$ for all n>0, and $a_1=16$