Question

Use the table to write an explicit rule and a recursive rule for the sequence.

Answer 1

Given the table:

n 0 1 2 3

f(n) 0.12 0.36 1.08 3.24

Let's write the explicit and recursive rule for the sequence.

Let's determine if the sequence is a geometric sequence.

`\begin{gathered} r=(3.24)/(1.08)=3 \\ \\ r=(1.08)/(0.36)=3 \\ \\ r=(0.36)/(0.12)=3 \end{gathered}`

The sequence has a common ratio of 3.

Therefore, it is a geometric sequence.

For the explicit formula of a geometric sequence, apply the formula:

`a_n=a_1r^(n-1)`

Where:

a1 is the first term = 0.36

r is the common ratio = 3

Hence, we have the explicit rule:

`f(n)=0.36(3)^(n-1)`

• Recursive rule:

Form the recursive rule of a sequence, we have:

`\begin{gathered} \begin{cases}{f(1)=0.36} \\ {} \\ {f(n)=3(f(n)-1);\text{ n>0}}\end{cases} \\ \\ \end{gathered}`

ANSWER:

• Explicit formula:

`f(n)=0.36(3)^(n-1)`

• Recursive formula:

`\begin{gathered} f(1)=0.36 \\ \\ f(n)=3(f(n)-1);\text{ n>0} \end{gathered}`