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

first term = 16

average ratio = 0.2

Answer 2

`a_(n+1)=0.2a_n` for all n>0,
`a_1=16`

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
`(a_(n+1))/(a_n)=r` forr all n>0.

To find r, note that

`(3.2)/(16)=(32)/(10(16))=(2)/(10)=(1)/(5)=0.2`

Similarly

`(0.64)/(3.2)=(64)/(10(32))=(1)/(5)=0.2`

`(0.128)/(0.64)=(1)/(5)=0.2`

Thus
`a_(n+1)=r a_n=(a_n)/(5)=0.2a_n` for all n>0, and
`a_1=16`